1987 Lesabre Occasionally won't start without battery dis/reconnect

I have a 1987 LeSabre that will not start sometimes and if I disconnect the battery and let it sit about five minutes and reconnect, it starts right back up. Then it seems to be fine for a few days, then won't start again. When it is trying not to start, it cranks, but when you let go of the key it dies again UNLESS the gas is continuously pushed down.
If you keep trying to start it without adding any gas, it seems like it cranks less and less each attempt?
There don't seem to be any other problems with this car at all and no real warning that this is going to happen. I can't even think of things that have been the same each time it happened. It has happened when car was warm, happened when cold. I guess it has happened after letting the car get below 1/4 tank and then refilling (sometime within a day of a fresh tank it does this, maybe?)

Check fuel pressure first, look at isc operation and tps functions next.