1997 Pontiac Grand Prix 97 Grand Prix intermittent start

Greetings,

This may be a bit lengthy so please bear with me. My wife's 97 GT (60K miles) has had an intermittent start problem for awhile now. 9 times out of ten, the car starts right up and runs fine but occasionally when you turn the key, the starter engages but won't spin the engine for a half a second and then starts right up. Other times, when you hit the start position on the key, nothing happens. You can usually turn the key off and back on a couple of times and the car starts right up or just hold it and around a half a second later the starter kicks in and works. When it's running, there are no warning lights, it doesn't overheat, runs fine and is very driveable.
Until this morning, this was only happening with a hot engine but now it will do it on a cold start as well.
It really acts like a dead battery or something dragging the battery down but even if the car sits for over a week, it will usually start right up.
Here is what I've done so far:
Ignition switch harness was replaced at about 40K miles due to a AC fan issue.
I've replaced the battery, the starter, redid the battery connections and replaced the main ignition relay as well. Alternator is putting 14.4 volts across the battery. I don't have a code scanner and am afraid to take it to get it scanned in case it strands me as I can't afford a tow. It doesn't act like the park switch as most of the time the starter will engage, just not spin the engine for half a second or so. Other times, the starter will not engage for a half second or so then fire right up.
I'm at wits end on this one. We can't drive the car as we don't trust it and it is really starting to cause problems as we need it. The car has been driven less than 100 miles in the last two years due to this. I'm almost thinking the battery is just not being charged enough but as I said, 9 times out of 10, it fires right up, even after sitting for a week or more.
What really scares me is the thought of something in the engine not letting it spin but as I said, the way it runs and drives gives no hint of a problem. The last time I took it out, it was twenty miles down the interstate and back home in traffic on back roads then I let it sit in the driveway in the sun ( 90 degress in Florida) running in drive with the AC on for an hour with no issues. I turned it off, waited a minute and then no start the first try and fired right up the second.
We've had three GTPs and this GT in the family but none of the others has done anything resembling this. My current 02 GTP has 160K miles and still runs like the day I bought it off of the showroom floor.
Any thoughts would be greatly appreciated. Thanks in advance.