The sound is probably from the power steering pump. If the sound gets louder when you turn the wheel, it's a sure bet. Check the fluid level, but with the age of the vehicle, the noise may be permanent unless the pump is replaced.
I have a 1994 Grand Marquis and the buzzing noise that I hear every once in awhile is the load levelers (rear shock system) filling with air. Maybe this is what you are hearing and it is a coinencedence that it stops when you accelerate My 1994 Grand Marquis does something like that...it's the air intake thing (press the gas pedal, it moves a valve at the end of the overhead cam). That valve can stick open slightly and hum. You may notice the car idles faster when it buzzes. This can be fixed by going over the edge of the valve with fine sand paper.
I would say you either have a rod knocking or lifter knocking. I know in the 3.8 litre motors they are known to knock alittle when you cold start them. Most Grand Ams, Grand Prixs are like this. My sister has a 2002 Grand Am that does this and the noise goes away once it's warmed up. I would also double check and make sure your oil is full. get your car to a garage for a def solution.
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