Being over 65, I have my own list of afflictions, and have friends of my age who have others. The most common ones relating to archery would be string arm shoulder or bow arm elbow, followed by arthritis problems in the fingers, etc. (The etc. means that we could go on discussing our various afflictions all day if we were so inclined, which we're not).
Generally, the problems we have have developed slowly over time due to overuse or aging joints and connective tissue that don't stretch and recover like they used to.
Your problem sounds different to me, more like an injury than inflammation from overuse. I wonder if you could be impacting a nerve in your hand with the bow handle? I know some people who ride bicycles have to be careful how they grip and weight the handlebars or they will damage the nerves in their hands. I don't think the symptoms you describe are common, at least I haven't heard about them before. I'm pretty sure they will go away after a while if you don't do things that would prolong them (like shooting the bow). But after they go away and you start shooting the bow again, you may want to experiment with a different pressure point of the bow against your hand. For example, try moving the pressure point more toward your thumb and away from the center of your hand, or the other way if it's already close to your thumb.
I would suggest going to a neurologist, but at this point I'm afraid it would be like the old joke where the patient tells the neurologist, "Doc, it hurts when I do this," and the neurologist responds, "Well, then don't do that. $100 please."