@richb
A little background to my comments - I have a colleague who is a classical guitarist with a dual faculty appointment at the Johns Hopkins/Peabody Preparatory and JH Neurology. She came to us (I'm part of an engineering group) to collaborate on some new technology to study how guitar related injuries happen, particular risk factors, mitigating aspects, etc. While I'm no longer
directly involved in the project, I picked up a reasonable amount of background info on this issue. Unfortunately normal orthopedic doctors are not necessarily equipped with all of the information they need to deal adequately with this type of problem, which is more akin to a sports medicine problem.
Long story short, the
primary and most common cause of these problems is excessive fretting hand pressure, and possibly too much squeeze of the pick since you're having issues in both hands. Rx can help defer the symptoms, but they won't address the causes and without a course correction you're headed towards some bad repetitive stress injuries. Same goes for stretching and other exercises. That's not to say they should be ignored, as stretching is a great habit to have. But it won't actually do more than buy you some time.
Your playing days are not coming to an end if you deal with the underlying source now. But if you try to continue treating it with medication, you are eventually going to be told that you can't play anymore and there's no coming back once you've hit that point. Given your age and presumed playing experience, you're going to have to unlearn some very severely embedded habits.
For a guitarist the easiest way to do this is to make some equipment changes to help force the issue. Move to lighter strings and the largest frets you can find - ideally a fully scalloped guitar but at least X-Jumbos if not. We're talking 9s or 8s on top of railroad tracks here. The goal here is that too much fretting pressure is going to push the guitar out of tune, and it will be much easier for you to correct what your hand is doing if you can hear the problem.
To deal with the right hand, my suggestion is that you head over to V-Picks or Gravity picks and buy something really fat. 3mm is probably a reasonable starting point but even bigger might be better. (Dunlop Big Stubby is a good choice if you want to trial this before spending acrylic money, but they're not really fat enough.) It's going to be harder to squeeze the hell out of these tubby picks. After that it's going to be about discipline - enough pressure to keep the pick in your hand, and not more than that.
Hope that helps. I don't mean to discount the suggestions above as there is good stuff in there, but it's not sufficient without changes to how you're playing the instrument. Some of these classical players are doing 6+ hours a day, and hand health isn't a game.