Shimmy at acceleration

I have a 2000 w/115k.When i start to accelerate it feels like the whole jeep shimmys from side to side it only lasts for a second or so then goes away.This has been going on since i bought the jeep with 92k on it.Recently it developed a vibration at about 65 MPH not sure if the two problems are related or not?Since the vib. started i have
1 New tires installed 215-75-15
2 replaced u-joints rear drive shaft and balanced.
3 Checked yoke at rear and tranny for play-None
4 Replaced fluid in rear w/lucas (no difference)
5 Pulled driveshaft and brake drums so i could spin axles freely and check for noise-seemed normal to my friend that knows a lot more about jeeps than i do.
6 driven at 65 MPH with rear driveshaft off and most of the vib. is gone.

Any help would be greatly appreciated

2 wheel or 4 wheel drive

4wd

Have you check you suspension for Torque steering. Something might be loose in the rear axle causing a shift when accelerating. Check the control arm bushings and track bar.

what about the front axle shafts...