Why is my beetle vibrating under acceleration?

So I'm new to this site...but I've gotten some good information from reading many posts here, in the past. I own a 2001 beetle sport 1.8l turbo...I've owned the car for approx 7 months, when I bought the car it had a bad exhaust (various leaks and rusted out areas) but the car always ran gr8. 2 days ago I had a new exhaust put on. I had driven about 20 mi when the car began to shake and vibrate under acceleration (most notably ) in 2nd and 3rd gear.. It felt like I had a loose wheel, but after driving it awhile I became aware it only happens when I accelerate and/or put a load on the engine.. Does anyone have any idea what this could be? I'm retired living on SS. Needless to say I don't have hundreds of dollars to throw at this....thank you

Check everything around the exhaust system that may have been loosened or removed during the work you had done. I've never owned a VW gasser but I can't imagine the exhaust system being routed much differently than a tdi.

Check the axle bolts that hold the inner cv joint to the transmission flange. There are six of them either hex head or triple square. It sounds like an axle or cv joint problem. Maybe the shop took an axle loose and didn't tighten it back up. Or take it back to the shop and have them check their work.

Your CV drive axles need attention. They are either:

A: replaced with junk Chaxles by a previous owner, in which case they are worth scrap and nothing more

B: they are still hopefully the original VAG drive axles and can be taken out, cleaned, lubed, rebooted, and put back into service

And if the above doesn't work, while you are working on B in oilhammer's post, consider swapping the inner CV joints on those axles. Check the races; if there are pits, then that is the source of the vibration, and swapping them will put unworn races in the place where the CV joints are stressed the most.

Cheers,

PH

Unless you do the work yourself just replace them all (inner and outer joints) The labor and down time to swap the joints from one side to the other is pretty much a wash with a one time replacement.
I've been successful with complete, new, (no doubt Chinese built for the China VW market), NOT remaufactured, complete axles from a national auto chain that has letters A -Z in their name. I've had crap US "remanufactured' axles that were not reassembled correctly and the new part was no more than clean grease in the worn joints purchased from the same chain at the same price.
New complete axles, or the same price for re-man, or more money for new joints onto your shafts, or re-install your used joints.... Decisions, decisions.