Congrats on your new car, and welcome to The Club. Seriously, these are great vehicles.

Engine mounts: A very common wear item. At that age, very reasonable that they'd be tired. I did mine many years ago - at about 200k miles. My car now has 400+k on it...

Tranny: Sure, yes. Diesels torque the heck out of their transmissions. You may need a new trans at some point, but...

As others have said: Check fluid first. Low level causing your symptoms is the first thing to check. With critically low level, car will absolutely not budge. Of course, locate the leak!

Another point: There is a very inexpensive (<$10?) bushing around the shift link to the transmission shift lever. It's a small PITA to install, but could easily account for all of your 'slipping out of gear' problems. I've gotta do this on my car, in fact.

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'83 300 SD
'05 E320 4Matic
'06 BMW 530 xi
'68 Triumph TR 250 - The only car I ever loved more than the Mercedes; who needs electricity, anyway? - Damn, why did I sell it?!
'59 Jaguar 3.4 'Le Chat Noir' - Damn, why did I sell it?!

It's difficult to make predictions, especially about the future.
- Niels Bohr