Q: When letting off the accelerator, the vehicle will jerk (it feels like a slight back and forth movement)

Hello. I think what we are looking for here are possible problems caused by a "torque-reversal" in the drivetrain. That is, when accelerating or maintaining road speed, the engine/transmission and all the drive links (driveshafts, differentials, transfer case, etc.) have force applied in one direction. When you let off the accelerator, the vehicle is now forcing all these bits to turn the engine - in the opposite direction. So any loose play in U-joints, CV joints, bearings, motor or transmission mounts, etc. catches this force. The loose motion, or "slop," can cause clanging, "thunk," or shifting of the engine/transmission on loose mounts. I would recommend having a your noise inspected by a professional mechanic, such as one from YourMechanic, to diagnose this problem, get an accurate assessment of damage and cost estimate for repairs.