That looks like what was in the house when I grew up...tossed that long ago!
Yes, I'm pretty sure you can buy a new stem for that. If the seat is worn, it might not work. If you're really lucky, you might just be able to replace the O-rings and gasket. You'd need someone to help you identify exactly what you have, and at over 60-years old, you might not find any easily available kits or a new stem. One of the pros will probably know for sure on that. Often, the best solution is to remove the stem and take it to a good local plumbing store.
The alternatives include tearing it out and installing a more modern valve, but it would be really close. That generally requires the use of a remodel plate, and I'm not sure that you have enough room between it and the tub spout - probably not. If that wouldn't work, if the new stem doesn't help, you might be able to cobble things up so that it doesn't leak, but then change the tub spout to one with a built-in diverter. If you had access from behind, you might be able to replace the diverter. Anything else I can think of would mean a bigger tear out, and remodel.
Note, depending on the extent of changes, it may trigger the requirement to update to the latest plumbing code which would mean a valve with anti-scald technology. Repairs generally do not trigger that, replacements do.