First, a bad steering stabilizer is never (!) the cause of Death Wobble. A heavy-duty steering stabilizer can help mask the symptoms of DW but it's never the cause or the real cure for the true underlying cause DW. And just replace it with an aftermarket steering stablizer if it's blown. Aftermarket steering stabilizers are bigger, less $$$ to buy, and will outlast the OE stabilzer which is downright puny.
DW Is most commonly caused by an imperfectly balanced tire and/or a track bar that is too loose and moving around. There can be other causes too but those two reasons... one, the other, or both in combination are a very common cause of DW. And I don't care if your tires were recently balanced, the TJ is far more suceptible to tire balance issues and the tire shops often only get them balanced to "good enough" which is never good enough for the TJ. Or perhaps one of the tires lost one of the tire balancing weights.
And when examining the track bar for looseness, grabbing it with your hands and trying to move it to see if it's loose is not going to be enough. It can be tight enough that you can't move it with your hands but still loose enough to allow DW to develop. The passenger-side mounting bolt needs to be tight (!) and the bushing must be in good condition to prevent movement. One way to see if it can move is to have a friend turn your steering back & forth while you watch the track bar and other front-end components to see if they are moving where they should not be moving. The trackbar should not be able to move even .0001 inch as the steering is turned back & forth.