'88 300ce engine dies while idling

[w124300ce]

Hi all,

I have a 88 300ce with about 100k on it. Recently the car started having problems starting as well as maintaining an idle. I had the car die on me in stopped traffic as well as in the mornings. Here is where I'm at:

It has trouble starting, engine runs for a couple of seconds at most, then dies.
It runs and drives however, when stopped, the idle descends slowly until it dies. By putting the car in neutral and giving it gas, this seems to fix the problem temporarily.
It is not an ambient temperature problem, I've had trouble starting in warm and cold weather.
It is not the spark-plugs, I just replaced them with Bosch Platinum 2 Plugs.

That being said (and after researching a tiny bit), I've narrowed it down to the fuel pump relay, fuel pump, fuel filter, oxygen sensor, engine coolant temperature (ECT) sensor. Please feel free to comment on what you believe the problem is, and if you've remedied this before.

Thanks, Paul