BMW Repair: Vibrations on Forward & Reverse Gear, bmw 318i e46, oil filler cap

Question
I've got a 2001 BMW 318i E46 model with automatic transmission that has so far done 68,750 miles. My problem is that the whole car vibrates excessively whenever I engage either the forward (D) or reverse (R) gear from stand-still position.  Same thing happens when I stop in traffic and the gear is in D position of course with the brake pedal down. However the vibrations diminish the moment the car starts moving.

The idling rpm is about 700 revs/min. This is when the car vibrates badly on D or R gears. The entire car vibrates and can be felt by all occupants irrespective whether the engine is cold or hot. Noticed that the vibrations tend to subside when I increase the rpm to above 800 revs/min by softly stepping on the gas pedal while keeping the brake pedal down. Turning on the A/C lessens the vibrations noticeably.

I don't know if its closely related but the 'yellow' oil level warning light immediately comes on whenever the car starts moving (i.e engine cold) and stays on for about 10 to 15 secs before it goes out. The red color indication disappears immediately after starting the car. I've checked my oil level on the dip stick and the level is perfect! Could the warning be that of the transmission oil? Shall appreciate your assistance in helping me identify the cause of these problems.

Thanks.

Answer
Al, faulty oil level sensors are very common an act EXACTLY as you describe as far as the yellow oil level indictator on the dash. Replace the sensor and do not reuse the old oil no matter how new it is. As far as the vibration it is very difficult to diagnose noises or vibrations over the internet, but open the oil filler cap with the engine idleing just a little bit (just crack it open) If the vibration changes pitch or gets worse the Crankcase vent valve (PCV)needs to be replaced. Also inspect all of the intake boots and hoses for leaks or cracks. This are the only things I can think of off hands. If none of these works out you might think about taking it in to the dealer or BMW specialty shop.
-Evan