I disagree with the "let it dry out and it will be fine" theory. Mine does that also, but only on left hand turns, and not all the time. Stepping on the brakes will make it stop and when I release the brakes, it starts blinking again rapidly.
I live in 29 Palms, which is a high desert and we get very little rain a year, and summer temps easily over 100 degrees. I think it's pretty dry if it ever got wet.
This is a known problems with TJ's (and Cherokees since they use the same assembely). The part you are refering to is known as the "multi-function switch" which is your blinkers and high/low beam selector. They go bad, don't know why. I know of no fix other than replacing it. I replaced mine (off Ebay, $16) and works OK. Still have the problem, though way less than I did before.
One trick is once it starts blinking rapidly, pull back slightly on the stalk (like you're going to flash the high beams). This will make it go to normal speed.
Hope this helps. Keep us posted on your results. Hopefully yours is the rare case where it dries out and is fine.