My 94' 850 just started not starting when it is cold out.

Whenever you try to start it, it runs, up to 1500 rpm and then stops. This can be repeated many times, always runs for 1-2 seconds and then stops. I have listened for the fuel pump, with the key on, not starting and hear it start then shut off, before turning the key to start. Lately, I have had to turn the key on/off (not engaging the starter) five times and on the sixth engaging the starter. After 5-6 times of this it starts and runs as if there were no problem. I can go anywhere, shop / etc and it will start right up, but if it sits out in the cold, I have to do the on/off trick. The last two times I noticed that after 2-3 times, there is a louder vibration-type of noise, right under the dash to the left of the steering wheel, close to where the fuel pump relay might be located (although it is on the engine side of the firewall). After this louder vibration comes and goes, the car starts.

Any ideas? Or just replace the fuel pump relay and go. I am about to try that this weekend.