This sounds like a problem associated with the drive shaft.
It is not a balance issue because it would not change with acceleration.
If it was just an alignment issue, you would hear the noise constantly. It is probably related to alignment but it is caused because something is moving out of alignment when you accelerate which means you have something loose. Could be transmission mount or differential arms.
Did someone replace the differential control arms? if so, did they install adjustable arms and realign the pinion angle? This should not go out of alignment unless there are adjustable arms which are set wrong.
Are the small quad shocks (differential dampers) in good shape? They could cause the problem if they go bad. They control differential pinion angle tilt when you accelerate hard.
Look above the drive shaft to see if the shaft is hitting the floor boards.
If you can not work on this yourself you should find another garage to do it. With a noise as bad as that, it should not be difficult for a good mechanic to solve.
If you want to try to fix it, check the drive shaft flange bolts to make sure they are tight. If it is, pull the shaft and rotate the joints while it is out to see if you feel rough bearings. When they assembled the shaft one of the roller bearings could have fallen out and be causing problems. I have seen cheap u-joints where the yoke is soft and quickly damaged by the roller bearings.
Check the flange on the differential for looseness while the shaft is out.
If you accelerate, can you see the gear shift lever moving? This would indicate a problem with the transmission or motor mount.