Ok, I hope I explain this right. I have an 09 - KJ and normally post there but I got an odd issue with my 02 KK so grab a beer and listen up.
I notice in the past couple of years, after getting some new wheels and tires from discount tire, that around 65mph, I would get what felt like an out of balanced tire. But after DCT checking it out, my "goto mechanic" for the KJ checked the front end and even had the dealer look it over during a free oil change. No one could find a problem. DCT, bless there heart, rotated, rebalanced my tires like 3 times. Still, didn't find the issue.
I have had the same issue with mine. Found out when I went back to the OEM wheels, that the aftermarket ones were these dual pattern things that would fit both Jeep/Dodge and Ford patterns. In this situation, the center relief was sized to fit Fords (about 3-4mm oversize)and the wheels were slightly out of concentric when they faced up to the hubs. Oh well, 158k on the old girl, it lived for 6 years with my daughter driving it, and 2 more with my youngest son driving it (I didn't buy it new BTW). So, one day, I decided to take off the DCT wheels and put my old KK tires on it with the big GoodYear duratracs still on the orig KK's rims. One is leaking air over time, but that's another story. SO, been driving around with these new treads and have just started noticing something. Say I'm driving in strait line at 15mph. I "goose" it. The rear feels like it wants to track over to the left, making the front kinda go to the right. Letting off, makes the front pull back to center, or the left. Do it enough and it feels like the back end is "wagging" every time you "goose" it.
So, driving down the road, normally, all is well. You can move the wheel back and forth and the car doesn't act like anything but a jeep. But going slow, feels like the booty end wants to go to the left and the it makes the front go to the right. I've check the engine mounts sitting still and racing the engine, motor doesn't move. Can't find anything BROKE at the tranny level. Not inspected the rear yet, this weekend maybe. If I do a jack rabbit start, really getting after it, you have to fight the steering a little to keep it in your lane. Nothing is rattling, bumping or thumping so if something is loose, I'll never know. Having had the car this long, I can say
I've never had the front/rear aligned. But this didn't start till I put new rims on from DCT which are offset a little so they stick out a little more then the OEM wheels.
Sorry for the long post but I'm sure some of you wise KJ owners might be able to point me in the right direction. FWIW, I just had a new power steering pump put on, along with some pullys and belts, hoses, etc.. about 700 bucks worth so I want my money back therefore, I've parked my 09 in the garage for now while I'm racking up the miles on the 02. Besides, I enjoy getting 15mpg vs 9. Thanks in advanced.
Try chocking the front wheels fore and aft. Have someone in the car, idling, with foot off the brake, and shift from R to D every few seconds while you watch the rear on each side. It's likely you will see one side more than the other, moving. This can confirm or deny where I suspect your problem is. (Rear Link)