How do you know it is the alternator?
You could buy a new one, fit it and still have the same problem.
You need to test the alternator and electrical system to isolate the fault - it could be internal to the alternator or it could be a broken wire between the alternator and battery or ignition controlled supply.
If this has been going on for 70 days, then it ...
I had similar symptoms on an old ML320. It turned out the be the camshaft position sensor. Probably coincidental that it happened when you changed the battery, but an electrical trauma like arcing when the battery was connected could send enough transients into the electrical system to end the life of a near gone CPS.
Pre-1999 E320 rings a few bells with me on transmission issues.
First thing to do - CHECK OIL!! make sure it's got enough of the stuff. This can help, if it's suspiciously low, this could indicate a leak.
Get a mech to check the TCU or ETC - these cars are known to leak oil / fluid into this control unit which can affect transmission.
The other thing is ...
Had the same issue in a VW Golf(1995 model). The car would shake violently when stopped a redlight. Tore the whole car apart and found out that my engine AND transmission mounts where very loose, got new mounts and tightened everything. So check the mounts, mechanic said that on older vehicles those tend to be the culprits.
Years ago when I used to inspect vehicles for a living. There was one thing that would cause such nasty fumes after 30 minutes or so. It was over spray from the anti-rust coating that sometimes would end up on exhaust parts and cause pungent sulfurous stench when hot. If it were an exhaust leak it would noticed a lot sooner and have a noise as well.