Dealing with big-time psoriatic arthritis. (Inflammatory, versus osteo "bone on bone".
First, and soonest, reduce recoil. Whatever you load, go to minimum weight, minimum velocity. Stop hammering your hands. Down-caliber the handguns you choose if you can.
If you can shoot weak-hand half the time, you can give each hand a weekend off. Or Gunfighter/Double-Deulist to do half the work each match.
If not already shooting .32s, try them.
Getting the hammers lowered (bent-down, super Blackhawk, bisley) and/or short-stroking can help.
BP minimum loads, especially with APP have rather soft-shove recoil versus smokeless slap.
Shotgun - feather-lights. Double? Punch it forward to unload, not yank backwards. If you can, try learning to shoot it weak- side.
Any non-cowboy high impact / vibration isn't helping. Learn to use computer mouse with the other hand.
Medical pros: Have them evaluate what is going on inside your hands.
Don't make things worse with wrong-stubborn.
Apply ice at the match when not shooting.