Tricep dips = pain
Hey JT, if your getting pain like that it's best to avoid. If your a body weight type of guy, and push ups feel ok, start with that and begin moving your hand position towards in until you are able to do a diamond push up. This works tri's nicely. Go slow and remember this can be a difficult move as all the force is centered over hands. Don't forget to work the opposing muscles, the biceps as well.
Bottom line though is talk to a sports medicine doc and physical therapist who can check your exercise form and recommend rehab exercises that will help. Good luck.