Yes, "tennis elbow" happens to slingers too! The only truly effective thing you can do about tennis elbow is cease the activity that caused it for awhile (six months to a year). There are several not-as-effective things you can do also.
If you are hooked on slinging, like I am, quitting is out of the question. That leaves two choices: learn to sling left-handed (I am right-handed), or learn some techniques that don't stress those muscles.
Personally, I have not the patience to switch hands (and my arm has not hurt enough to force the issue). I have had pretty good results from making sure I warm up properly before slinging, and from learning how to sling with the "unbendable arm" that was discussed in
another topic
. I think reading the discussion will be useful, but I'll summarize it here.
The "unbendable arm" is a technique taken from aikido, according to SEB. You don't hold your arm rigid, you hold it stiff but relaxed (and there's a couple of mind tricks to achieve this; SEB suggested imagining your arm is a firehose, with water rushing through it full-force). Holding it rigid means your muscles are tense, which means that opposing muscle groups are pulling on the bones, which means that any force applied to your arm unbalances the forces and your arm bends suddenly. Holding it relaxed means your muscles are not tensed, which means they are free to react instantly to any force applied to your arm. In short, you hold your arm as rigid as you can, and I can bend it by something as simple as popping you on the inside of the elbow; you hold your arm relaxed, and it is unbendable.
If you learn the "unbendable arm" technique (and it is surprisingly easy to learn, I don't even have any martial arts training and it only took me an evening) and if you use it consistently (that's harder, I keep forgetting to do it) then you will be able to sling
in moderation without stressing those forearm muscles and tearing them away from the bone more than you already have.
Of course, if you have a continuing problem with tennis elbow, then you never have really let the muscles heal. It does not work, just to wait until your arm does not hurt any more. The damage is all where the muscles attach to the bone, and the blood supply is not good at those points (maximum strength in tendons and ligaments is not achieved by perforating them with blood vessels). It does take six months to a year for the damage to really heal. You may need an extended period of not slinging, or you may even need surgery -- followed again by an extended period of not slinging.
Hope this helps. I've had some success when I remember to do as I have just preached, but if I forget even a few times, I'm hurting for several days.