HelloCustomer and welcome to JustAnswer.com!
Based on your symptoms, I kind of have to agree with your mechanic. Most likely you've either got a bad axle, a bad motor mount, or both.
A bad motor mount is simple enough to diagnose. You simply "brake
torque" the engine
(with a wheel chocked and while standing on the brake pedal, you put it in gear and hit the gas pedal, while someone else watches for excessive engine movement) in both forward and reverse gears. From this a good technician should be able to identify a bad engine mount.
In order to properly identify a bad axle joint, sometimes the axle must be removed from the vehicle and the joints moved by hand checking for excessive binding or play in the joint. It is not possible to 100% identify an axle as good just by inspecting it on the vehicle.
Sometimes an axle joint can go bad just from normal wear and tear, but a bad motor mount can accelerate this wear by increasing the engine angle, thus putting the axle in a bind during acceleration.
A rhythmic "clunk,clunk,clunk" that increases with vehicle speed during acceleration, especially during turns, is a classic symptom of a bad axle joint. It's often less expensive to replace the entire axle than to replace just the joints, as there is less labor involved (no need to disassemble the axle once off of the vehicle).
I hope this helps.. Good luck!