OK, I've been having this problem for years now at this point, but I'm finally at a point that I can take time to address it. It originally felt like my engine was flopping around a while back, so I got new front and rear trans mounts plus ES red poly inserts to stiffen things up past stock. The old mounts were definitely torn a bit. But even directly after installation, the engine still felt like it was flopping/twisting around. Especially starting off in 1st, letting off the gas in gear in any way but super smoothly, or accelerating moderately/hard over a pothole, the car just bounces back and forth like the front and rear engine mounts are shredded. Still have the interior shake/rumble that arrived with the poly rear mount, so it can't be worn, and the front looks fine. Tried tightening all of the bolts, but they were fine, and there's no sign of things being loose enough to cause wear on the mounting locations.

So I'm kinda stumped. Any clues? Could the clutch springs be worn enough that they'd cause a similar issue? Not sure how many miles are on the clutch, but I've put on about 70K since getting the car and it still grips fine. I also replaced the timing chain side engine mount for some reason or another (I forgot why), but the engine was definitely raised up a bit with the new mount installed (no poly). The only old one is the top/side of the trans housing, so things are in reasonable shape concerning mounts.

 

Hmmm...so there are 4 mounts, you have replaced all 4?

If you have, tell me this (you will need an assistant for this): start car, sit in park. Lift hood. Have assistant watch engine running. Now keeping your foot on the brake, put it into gear. Does the assistant see the engine lurch forward? Now put it in reverse, still keeping your foot on the brake. Does the assistant see the engine rock the other way? Is so, it's likely still a motor mount.

A sensation like you are describing could be old shocks. Or maybe you have other old suspension pieces. Have you inspect the control arm bushings all around, that can make a weird sensation when completely broken.