I have had significant issues both Medial & Lateral Epicondilitis in both elbows from years or rock climbing, so I feel your pain. I fired mine back up a few of years back when i got into Crossfit and started doing lots of PullUps/PushUps and poor clean technique. I did pick up some Elbow Sleeves from Mark Bell and they help keep your elbows warm and they feel good ... but as they won't solve the problem. The surgeons were gungho to carve my elbows up, but I opted not to go that route as my problems were symmetrical enough that it didn't make me believe that it would be more than a short term fix.
After resting them and ultimately doing injections in both to help get them under control, I found that it was a function of thoracic spine mobility and tight triceps. FWIW, my triceps felt 100% fine, but I found that mashing the heck out of them over a barbell along with voodoo flossing was like magic and I continue to do these things regularly and I am pain free for big volume of clean, pullups, pushups etc ...
FWIW ... I'm 53.