Hi all. Finally got this one figured out and decided to post what I found.
There were two major problems.
First, I replaced the MAF sensor, but that in and of itself did not take care of the problem.
Second, I found that a mouse (I assume it was a mouse) had chewed on some wires and stripped the insulation off the bundle of wires which are located below the air cleaner housing on the driver side of the car. Sorry but I am not sure what this "bundle of wires" is called. Some of the wires were bare and touching one another.
The combination of repairing the wires and replacing the MAF sensor seems to have taken care of the problem. I let the car run for about an hour on two different occasions. Decided to take it down the road. Have only put about 10 miles on it, but things appear to be working correctly.
The fuel system still does not hold pressure when the fuel pump is not running, but while the car is running, all appears to be OK. I will need to address this next because there are 270,000 miles on the care and the fuel pump has never been replaced.
Let me ask another quick question or two if I may.
Is this model of Grand Prix (1998 GT, four door, 3.8L non-superchared engine) one of the "lucky models" which allows access to the fuel pump thru the trunk, or will I need to remove the gas tank to replace the fuel pump?
Thanks again for all of your replies. You have all been very helpful and I will keep an eye on this forum and offer assistance whenever I can.
John.