I have exactly the same problem. I have just got back into paddling after many years off and I have developed tennis elbow in the right elbow and golfers elbow in the left. I have been to the physiotherapist and he has given me exercises to do but sometimes the elbows are just too painful for me to be able to do them. I have tried anti-inflamatories without success. From what I can figure out, the only way to get better is to stop paddling, but I have a race in April that I need to be fit for. Strange thing is that the elbows are reasonably OK when paddling, it just hurts afterwards and I cant do a proper gym session. Would love to hear if anyone has had this problem and has fixed it.

I have the same problem in my left elbow and have been for three sessions at the physio.  Highly recommend Nikki Akermann in the ABSA Centre in Cape Town.  The elbow is still stiff but a lot better, though I should probably be resting it even more than I have (that can wait till winter).  I've also changed my stroke to concentrate more on shoulders and body rather than arms (and especially elbows).