Yes, you are pushing at a low cadence on the climbs and that can cause cramping.

It becomes a balancing act with gearing -- suffering through relatively short periods during the ride, and having the broad range of gears you need for later on.

You almost sound disappointed that you don't think your leg strength is up to scratch. But really, in my book, you are doing a fine job running with the big dogs in the A rides. More riding and maybe some weight work on the quads with lunges and squats might be useful.