We've got some terminology problems here that are making it difficult to understand what you're experiencing.

If you turn the key and the starter engages the engine so that it turns, that is cranking or turning over. If the engine does not catch and run on its own, that's cranking but not firing. When we talk about the engine "turning over", that means it's being cranked by the starter. From what you ask, it's not possible to tell if the starter makes noise but doesn't engage the engine so the engine doesn't turn, or if the starter is turning the engine but it's not firing, or if the starter was silent (or made just a click) until you replaced it.

I am guessing your problem is that once the engine has warmed up, it will crank and crank and crank but not fire and run. This is a pretty common issue that has nothing to do with the starter. It's a fuel-related problem known as percolation that happens when things get hot under the hood. Never pump the gas pedal while trying to start a hot engine! The correct starting procedures are as follows:

Cold engine (sitting more than a couple of hours): press the gas pedal to the floor and then take your foot off it and do not touch it again. Turn the key until the engine starts. If the engine starts and then stalls, press and release the gas once more and then turn the key.

Warm engine: press the gas pedal about 1/3 of the way down and hold it steady there without pumping. Turn the key until the engine starts.

Hot weather and worn carburetor parts, together with modern alcohol-contaminated gasoline, aggravate hot-start issues. Do the Fuel line mod, and go through the carburetor; Carburetor operation and repair manuals and links to training movies and carb repair/modification threads are posted here for free download. Make sure you've got the thick gasket (about 3/8") between the carburetor and the intake manifold, not the thin one (about 1/8").

And before any of this, you will want to get and read the three books in this thread as quickly as you can.

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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 suspicion you have is correct, it is not firing. Thanks! Is there a gasoline or gasoline additive I ought to be using on a regular?

There is no special gasoline or additive you should be using. Get those books and do those mods.

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

This will be one of the more stupid things someone will have asked you (Dan).

Is it fine for me to not pump the gas pedal before turning the key for a cold start up?

Thanks.

Engines with a carburetor and some type of choke will have a start-up sequence that will ensure fast starts with-out a lot of peddle "pumping".

Most SL6 engine with a properly working carb / choke will take one push of the gas peddle, to "set" (close) the choke, then the engine will start, when the key is turned and the engine "cranks-over".

Here is the funny thing... I have a bunch of SL6 A-Body cars, and each one seems to start a little differently.
My Dacuda starts like described above: 1 pump, turn key, it starts and fast idles, rev the engine once, the choke drops-down to a medium idle choke setting, put in gear & drive away.

My 66 Dart wagon, it does all the above but once it starts and fast idles for a moment, it stalls. Pump the gas peddle once more, it starts again and does the "kick-down, drive away, no other issues" like it should.

Our 64 Dart, set the choke & crank for a moment, no start. Stop cranking, push the gas one more time, turn key& crank again, it starts and does the fast idle, kick-down & warm-up, like it should...

If your vehicle takes a lot of pumping and "fussing" to start, keep running and warm-up... then it is likely that the choke, carb or fuel system needs some attention.
DD

It won't start or stay running that way. Follow the "cold engine" instructions I posted above.

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