I've been through a rough year of cortisone withdraws which has drained a lot of my bones of calcium and I have a lot of arthritis. Mostly only in my fingers and hands now.
Because of my medical situation I'm not supposed to take any kind of anti-inflammatory (NSAIDs) either.
A few things helped me. Getting amazingly warmed up in every joint. At one point I couldn't really squat because the bar on my rear delts was too painfull and my wrists and fingers wouldn't turn to grip the bar. My hands still don't.
My elbows were tender to the touch for months and I got a pair of elbow sleeves from APT. They aren't thick like my Rehbands are but they are extra long (good for my monkey arms), size Large fits my arms (19" cold, easily), andthey really keep the heat in well. I always put Ben-Gay (generic works fine) on all joints before I warm-up even. Also started wearing my wrist wraps (also APT) to make sure that the pain wasn't being directed from my wrists.
A lot of it could be from your exercise selection or the % of your max that you are constantly pressing. Can you give us an idea on these?