What camshaft should I use?

I suspect it. I saw a few pushrods spinning but I need to double check. I know that no matter what I do to the car it will have a rhythmic “pop-pop” out of the tail pipe, you can hear an audible drop in engine speed that matches the misfire out of the exhaust, and looking at the engine you can see it shaking when running.

Here is the dead giveaway if it is an exhaust lobe. Remove the air cleaner. Start the engine. Drop it in drive (I assume auto). Hold the brake firm. Load the engine up against the converter. If you get a rhythmic POP out of the carburetor, you have "something" holding the exhaust valve closed. Whether it's a wiped lobe, bent pushrod, broken rocker......or "whatever".