I have the same problem with my elbows for different reasons I think. My more serious problem is the top of the forearm or tennis elbow, more so than the inside of the elbow which I think is what you are describing. I have toyed with different grips too and found that if I move one to alleviate the pain on the forearm and focus on that I start to get tightness on the inside. For me the best path was to switch to chin ups instead of pronated or standard pullups and decrease repetitions while increasing weight.
Since the tendons around my elbow joint are all giving me nagging issues I have done this on many exercises to reduce the amount of use. I am also working on grip and forearm strength a little more which I hope will relieve the stress around the joint.