Probably because the ball your using is too heavy for you.
Why do your fingers feel weird/weak after bowling?
Mine usually snap off, I get new ones the next day.
They have been in the bowling ball.
Because of holding long time the ball of bowling, then they feel weak and weird, it's normal, as we have used them for a long time to hold it, there are some people had cramp after playing it as well.
Well, I have only one reason for that - your fingers need practice. The same about exercising. The very first day is the hardest, your muscles are not trained well and ache the other day. So I recommend you to train your hands and fingers everytime if you do not want to feel weakness.