Adam, a 4 cylinder 12V coil will work just fine. Do you know what type of distributor you have? A pic may help.
If you suspect the pump due to the black ooze get another one and before mounting and playing install a fuel pressure gage between the polishing filter and the carb. I would suggest fresh fuel lines when doing this, often black ooze is ethanol breaking down old style fuel line.
Also a coil from a 4 cyl V-dub would work just fine. Do check the points and integrity of the cap. Also give the C-adv a twist and lube if necessary. And yes a coil will heat up!
When testing if she ever shuts down check for spark~~BE PREPARED~~immediately. No spark its ign and probably the coil.
You will already know about fuel delivery as you now have a pressure gage you checked while starting and/or running.
When the engine dies how does it die, stumbling to a stop or just quits like you shut her down?
You may want to start a fresh thread for your issues.
Dave Neptune