You may easily have just a bad alternator. Convention is to remove it and have it tested. Usually free.

I would just get a used alternator from a salvage yard before buying a cheap chain store alternator. If your altenator tests bad. Cars do not get scrapped because the altenator is bad. Chain stores can provide properly rebuilt altenators reconditioned by the original manufactuer but they will cost you.

The chain stores tend to buy so called rebuilt alternators from Mexico. They are hit and miss for too many being bad. Or having a very short lifespan.

Remember starting a car and pulling a battery cable is not a good ideal. Nor is boosting one. You may get away with it especially on these old cars. When you do not on some cars. It can really cost you dollars.