With switch lever in Right position,
the light on the instrument panel does not light up at all.
The front and rear right signals blink and you can hear the flasher unit.
Everything works fine except the light on the instrument panel doesn't blink.

With the switch in the Left position,
the light on the instrument panel does blink,
but only the left rear signal blinks.
The flasher unit sounds the same.
I removed the bulb and it the filaments were good, but I replaced it anyway and cleaned the socket with contact cleaner and a piece of scrub pad.
No change, though.

Fuses are OK.

For left turn, the instrument indicator works, but the front left signal doesn't.

(Note: the 63 instrument panel has only one light that lights up for right or left)

I checked, and the right front turn signal socket is grounded properly.

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63 Signet / 10 inch front drums / weber 2bbl / HEI / 2.76 rear end

You need a new turn signal switch (see here); yours is worn out. Odds are pretty good your brake lights don't work consistently, too.

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一期一会
Too many people who were born on third base actually believe they've hit a triple.

You can confirm or eliminate the switch being the issue for the left front by connecting a voltmeter or 12v test light to the 2 light green wires coming out of one contact in the turn signal connector; one of those 2 light green wires wire goes to the left front blinker lamp. See if there is on-off voltage at those 2 wires with the switch in the left position. If not, then NG on the switch. If the on-off voltage is there, then the wiring, the firewall connector, or front left socket/ground is the issue. I suspect you will find that the voltage is indeed there, as the 2nd light green wire feeds the signal indicator in the instrument cluster, which you report as working. So the issue is likely in the light green wire going up front. (You can also check for the on-off voltage on the dark green wire which goes to the left rear blinker/brake filament as a double check.)

Similarly there are 2 tan wires from 1 contact in the signal switch connector; one feeds the right front signal and the other feeds the indicator in the cluster. If the right front signal works, then there should be on-off voltage from the signal connector on the 2 tan wires, since you report that the right front blinker works. If this is the case, then the issue is in the tan wire to the instrument cluster, the cluster connector, or the indicator lamp.

You can see it all in the electrical diagram (2 pages) for your car that you can download from mymopar.com

BTW, I have not figured out how the one indicator is fed by 2 wires; maybe they are paralleled inside the cluster in a way to prevent crossfeeding the 12v or the indicator bulb has dual filaments. My '62 Dart is the same way so maybe someday I will know!

Edit to update: Well that day is today.... I looked at the '62 factory service manual and the schematic for the '62 Dodge and Lancer both shows what looks to be a dual filament turn signal indicator bulb in the instrument cluster. But if I look at the bulb list in the FSM, it shows a number 57 which is a single filament bulb...! So the FSM bulb list is wrong. It could be that you have a single filament blub in your cluster and that is why the turn indicator only work on one side....Look at the cluster turn signal socket and see if it has 2 wires to the base (in addition to a ground). I don't yet know what the correct dual filament bulb would be.

Just to confuse you, I believe the bulb is fed from the left and right positives for the front turn signals. The single filament would ground through the unpowered sides front bulb. Simple, really.
I believe Dan could confirm either solution.

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Mattelderca
78 NYB (gone now), two S series, three old Snow Cruisers and a Doo.

Not confusing....probably illuminating (pun intended)! Kinda makes sense as I presume the outside filament resistance is a lot lower than the cluster indicator filament resistance. What is confusing, however, is that my '62 FSM schematic shows 2 wire connections and a ground too, so ...... I'll be heading outside with a voltmeter shortly!

The turn signal lamp is the same as the others.
Just two contacts on pcb for each.

It's the upper middle one that is off-center.

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63 Signet / 10 inch front drums / weber 2bbl / HEI / 2.76 rear end

That makes sense.

Yep, you look to be right in the indicator feed. There is a very small voltage on-off on the opposite filament on my '62, only about 15 mV. A pretty elegant solution, IMO.

This would explain the front blinker working on one side but not indicator, and the indicator working on the other side but not the blinker. Sounds like the wiring to the left side blinker is broken or the left blub or socket is bad. One problem can be causing both symptoms.

Now I am getting confused.
Hopefully, tonight I will have time and it won't rain again and I will check the two light green wires for on-off voltage.

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63 Signet / 10 inch front drums / weber 2bbl / HEI / 2.76 rear end

No, you can't, not reliably. When the switches wear out, they create a bunch of intermittent/random problems. Sometimes the shuttles make contact, sometimes they make partial contact, sometimes they make incorrect contact, and sometimes they make no contact. You can chase your tail for days, weeks, or months thinking you've "eliminated" the switch as the cause because it happened to make contact the time you happened to check.
No, it's a single-filament bulb, and that's what's shown in the schematic. It works for a left or a right turn alike.
N/A, it wouldn't.

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一期一会
Too many people who were born on third base actually believe they've hit a triple.

l stared at the schematic for awhile.
How could the indicator lamp on the dash blink but not the outside left front blinker, when both light green wires come off the same terminal on the turn signal connector?
It occurred to me that the light green wire that goes to left front blinker lamp has to go through the bulkhead connector and the other one goes directly to the dash.

I cleaned the terminals on the bulkhead connector, and it works now.

Thanks for the ideas, all.

Or maybe it just randomly started to work, and had nothing to do with cleaning the bulkhead connector.

Could a bad connection at #8 terminal on the big bulkhead connector cause the original symptoms?

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63 Signet / 10 inch front drums / weber 2bbl / HEI / 2.76 rear end

Any bad connection can cause problems with lighting and other electrical circuits, particularly ground side of those circuits.

I have to agree with Dan, that directional switch is an intersection of all but headlight & interior lighting circuits, and once it gets to be 50 years old or so, it reaches its cycles to failure limit, and needs to be replaced.

It would be a good idea to go through all lighting ground connections, bulkhead connector, and bulb sockets making sure all are clean & tight with no voltage drop across them. You mat find that replacing some of the brass connections is necessary sometimes it's a faster more reliable method of fixing a bad connection; for selection of various connectors and housings one place to get them is here and here if needed.

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67' Dart GT Convertible; the old Chrysler Corp.
82' LeBaron Convertible; the new Chrysler Corp
07' 300 C AWD; Now by Fiat, the old new Chrysler LLC

That is correct. Current flows thru the single-filament pilot bulb in one direction for a LH turn, and in the other direction for a RH turn.

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一期一会
Too many people who were born on third base actually believe they've hit a triple.

Yes, the #8 bulkhead terminal connection being dirty/corroded would make perfect sense as to why left front blinker did not work at all and why the turn signal indicator did not work when the right trun signal was enabled.

And I'll disagree on the troubleshooting process a bit here; if there IS on-off voltage out of the blinker connector, and the front blinker light is not working at that point in time, then there is obviously a wiring-to-the-blinker or bulb/socket issue at that point in time. That appears to have been the issue in this case; using diagnosis rather than just throwing a part at it.

That procedure diagnoses a non-switch problem that needs to be fixed. It does not preclude that the blinker switch could be erratic at another time. If you fix the wiring and are concerned that the switch is erratic, then either:
-rig up voltmeter connection onto the switch connector and check it for consistent on-off voltage each time you enable the turn signal while driving around.
- or just throw a new part at it if you prefer

I won't gainsay the thoughts expressed about issues with 50 y.o. switches. But I'll still start with actual diagnosis.

Well, I still am putting a new turn signal switch on my Need list.

But I need some major exhaust system work now.

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63 Signet / 10 inch front drums / weber 2bbl / HEI / 2.76 rear end

Two different solutions seemed to fix the problem:

One time, I disconnected the battery, cleaned a bulkhead connection, and then it tested OK.

Then another time, the signals didn't work one morning.
Parked it for an hour,
did nothing,
and two hours later left and right both worked correctly.

It is the turn signal switch.

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63 Signet / 10 inch front drums / weber 2bbl / HEI / 2.76 rear end