Your problem is the automatic transmission relay, which is what makes the car start in only park or neutral. On my A4, it is relay #4, counting from the left on the front relay panel under the dash. Your A6 may be different, but the plastic fuse cover should have a diagram that shows the relay location.

Remove the cover under the dash. On the A4, one bolt is behind the fuse cove panel, the other three are accessible after removing plastic covers. If you are not sure which relay, try to start the car, and feel for the buzzing relay housing. Unplug the relay (pull it straight out), then un-clip the plastic cover with a small flat blade screwdriver. If you have 1000 super-fine aluminum oxide sandpaper, use that to polish the contacts. Then, wet a white paper business card with alcohol and rub between the contacts. You will see a black residue until the contacts are cleaned. Plug it back in, and the buzzing/starting problems should be gone.

I hope my suggestion helped, but it was a little too definite. Excessive resistance in the ignition switch or transmission switch (called the multi-function switch) could cause the buzzing too. In addition to that, in the transmission relay, the little leaf-spring contact might be bent, and not giving full force to the contacts when they are closed by the magnet. If you take the cover off, look carefully under a magnifying glass at the flat leaf-spring. If in doubt, bend it a little so the moving and fixed contacts are closer.

If this doesn't help with the buzzing, I would suspect the ignition switch next. They are inexpensive, but some work to replace.

I changed my mind, because the A4 did the buzzing again last night. The voltage at the relay was only about 5V while it buzzed, then was about 12V when the engine started. That leaves the ignition switch as about the only thing it could be. I pulled the switch and removed its cover. The starter contacts (contact 50) were burned-looking. I cleaned these up with some #600 aluminum oxide paper and wiped dry.

So far the car is starting normally, if anything changes I will let you know.