Accelerates rough/weak acceleration 2002Jeep GC

could be a engine mount, any engine setting out of line will add more load to the engine mount movement, air leak in manifold, bad spark. it could als be something as simple as a thrown wheel balance or the balance on drive shaft balancer is out of sync, or missing.

the overheat problem is the thermostat

I have a 94, and mine ran rough until I had a new computer control chip put in. The dealer doesnt sell then anymore, so you would need to get a reconditioned one (is that the right word?) from an auto parts store. I had to get this done twice because the first one was bad so they swapped it out. There was one more thing the former owner did to fix a rough run situation, I will post it when I find the receipt. Also, it smoothed out even more when I changed out the motor mounts. WAY less vibration. I love to drive this car now!

Sounds like you need new engine mounts

Personally, the first place I would be checking is your air filter.

Rough Running Engine - James on Engines #2 - Bell Performance

https://www.bellperformance.com/blog/rough-running-engine

Apr 21, 2015 - When diagnosing an engine with a rough idle, there are certain things that must be ... If the problem does not reoccur, the codes will be erased by the computer. ... when accelerating the vehicle, which causes a higher fuel demand. .... it was water in gas...added added Gas LineAntifreeze/Water Remover.

buy a bottle or two of gas line anti freeze, it absorbs any water that may be in your fuel. add one bottle now and one the next time you fill up. if it is water as i suspect you will notice results almost immediately.

check the drive shaft between the intermediate transmission and the front transmission for miss alignement and defectivness of the telescopics system of this shaft. we had about the same problem on our 1996 same model...

I would venture to say your exhaust is hitting your frame, or, you may need new motor mounts. The exhaust is what I would lean toward. If you can imagine your exhaust sitting under your vehicle, and the fact that the vehicle has a little age, the motor mounts probably give a little, causing the motor to move slightly on acceleration and deceleration, thus your exhaust is going to move, and may be vibrating on whatever its touching. Good luck.

It's hard to understand when you say it feels like a manual shift. Noise? Feel? I do know that the 87 auto trans does have a weak 1st to 2nd shift drum (as explained to me by my mechanic) that makes a grinding, elongated shift under heavy acceleration when it starts to wear. Under normal acceleration, 1st to 2nd shifts were normal (crisp, immediate). Since that day 8 years ago, I've avoided starting off to quickly, and so far have saved myself a tranny overhaul.