I'll try to make this short skipping a whole story. 65 valiant.
Radio shuts off and turns back on when driving. And been getting worse. Currently it shuts off at idle or low speeds when headlights are on and I flick the turn signal. It only shuts off and comes back on on the first blink. The others don't affect it until I turn blinker off and then back on. It doesnt do it as often on freeway speeds. Sometimes it will though.
I have a new battery, also just changed out the ignition switch as I thought it could be a problem.
(Originally battery went bad. Enough juice to keep dash lights on but key wouldnt do anything , but a screwdriver to the starter would crank.) Also some times when I would jiggle the key whilet driving the radio would shut off and back on.

So I think I may have a grounding issue or alternator possibly? Leaning toward's a ground since screwdriver to starter works whem volts are low and bypasses grounds when using key.

Check the bulkhead connector, and connections on the back of the ampmeter

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Charrlie_S
65 Valiant 100 2dr post 170 turbo
66 Valiant Signet 170 nitrous
64 Valiant Signet
64 Valiant 4dr 170
64 Valiant 4dr 225

Years ago my first Dart, a '67 270 273V8, I discovered that with engine ignition off, radio on, brakes applied (brake lights lit), and any one directional light flashing the radio would come on and off in lock step with blinker... Somehow electrons were back feeding through the harness to power the radio. I suspect something similar is happening with your car.

I would disconnect, clean and remake ground connections between battery to block, head to fire wall, and make sure radio has a good ground to dashboard.

Look for any wire that may be chafed, or broken within its insulation, or has a corroded connector that feeds the radio, ignition switch, and directional signal switch. Look for a fuse holder in line with radio feed that may have a bad connection or fuse. Somewhere there is an intermittent connection activated by vibration.

One way to check is to wiggle wires under dash when radio's switch is on but radio is not getting any power, and hopefully it will energize once the bad wire or connection is fondled... Yes fondled. Be genital, one customer at a time, and you may quickly be rewarded.

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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

Check your grounds.

Weird things happen with bad grounds........

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Ed
64 Valiant 225 / 904 / 42:1 manual steering / 9" drum brakes