I suggest some finger / grip specific type of exercises. With all the PPE players use growing up (my son uses a glove insert), if the hands themselves aren't conditioned they can and do get more swollen / sore than if not. Definitely agree with catching in the pocket etc and everyone's comments above. Guitar players have a spring loaded device that isolates the four fingers, using the thumb to stabilize it that literally works out each finger individually against spring tension. The old black handle with the loop of 1/8" steel that you sit there distractedly squeezing over and over...that type of thing.
This is my first post and I am by no means an expert. I am the father of a great 16 y/o catcher and I am literally looking at this as if we were standing in a room together discussing the issue about one of our sons. Only a suggestion. maybe not the cure itself, but perhaps in conjunction with everything else above???
God Bless America!! I am really happy to be communicating about baseball with like minded individuals 
Scott