Ice has done wonders for my aching elbows.
Also, you might a better response from a different anti-inflammatory. I tried Celebrex and I never noticed a difference. With Bextra, I can tell it helps.
I imagine that you're experiencing tendonitis. Some people push through it and it goes away. Some people train around it and it goes away. I recently had a bout with biceps tendonitis and I tried all of the above for weeks. Unfortunately, I had to call it quits in the gym for 4 weeks. I didn't touch a weight for what seemed like a lifetime. I can tell you that I didn't loose any bodyweight or size and my strength was still up when I returned. Hope that helps. It may or may not pass.
I also did 6 days of a Methylprednisolone prescription. It knocked out the biceps tendonitis, but I still have slight pain behind one of my elbows that I keep ice on. Another tip is to be preventative once it heals. Ice it after each workout to keep the inflammation down and from coming back.
Good luck with it. I hope it passes quickly.