Well it probably has nothing to do with the Tylenol, first off.
Whatever muscle relaxant you are taking, that is what may be causing it. While rock-hard erections is not a typical side effect of muscle relaxants, it may be possible it is helping you get them since, as you mentioned, cock has smooth muscle tissue in it and maybe somehow this medicine is allowing additional blood into your corpus cavernosum.
Or, like someone else alluded to, the absence of pain is allowing you to get more horny during those times.
In the meantime, sorry to hear about your jaw pain. A girlfriend of mine suffered from TMJ horribly and what made it extra difficult for her was the extreme lack of sympathy and understanding from others. People expected her to be able to sing, to be able to chew, and dentists never understood why she couldn't position her mouth the way they wanted. When you tell someone you have TMJ, it's as though they completely forget that fact mere seconds later.