When I was doing more "rock" gigs that I had to use a pick for I would have the same problem after 30-45 minutes. Darn near impossible to hold the pick. I saw several doctors about it, most of which wanted to operate (CTS), which I was not really up for doing...especially in the middle of extensive touring.
What did I do ? For rock gigs I switched to a thumb pick. It takes far less pressure to hold on to them and they're easy to make the transition to. I still hold them in a similar fashion to a regular pick, but due to needing very little pressure to grip them, I can play for hours with no problems. I use the National ones typically, though Dunlops are good for a different tone. (IIRC, Nick Lowe used a thumb pick in the 70's)
I rarely use picks these days, as my gigs don't call for them as often, but when I need one, the thumb pick is what saves me from excruciating pain.
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I used to use thumbpicks as well, but didn't find I needed to grip them at all, there's no need to grip them like a plectrum. I still use one on very rare occasions, you get some really good string attack wth them
As to cramps, I sometimes get them as well, possibly this is due to dehydration so you need to drink plenty of water.