Have you noticed a drop in MPG?
Could be a leaky injector or Fuel pressure regulator.
Cold engine can run richer and won't mind, and any fuel in the intake would have evaporated, O2 sensors are not online for the first 5 minutes as engine warms up.
Warm engine can't start well rich, and O2 sensors should compensate for small leak once started, so engine would run fine, just lower MPG
Fuel pressure gauge would ID this, '97 should be 35-40psi running, and not drop below 28psi with engine/key off, check it after 10 minutes of key off.
You could also pull off the air plenum(big air tube) on the intake, warmed up engine, open the throttle and smell for gas, there should be no gas smell.
When you turn off a fuel injected engine, the injectors shut down with the key, then engine stops rotating a few seconds after that, so there would be no gas/fuel in the intake it would have been sucked out by engine rotating.
Could be Coil Pack is starting to go, once it heats up it has a weaker spark, which will still fire running engine, but after shut down, while still warm it takes a few on/off triggers to get/build enough spark.
Starter motor drawing too many amp warm, do you notice it cranks slower warm?
IAC(idle air control) valve sticking closed.
On cranking/startup the IAC valve should open all the way, so RPMs should go up to 1,500, then if engine is cold RPMs will drop to about 1,100
If warm RPMs should drop to about 750.
If IAC valve isn't opening all the way on cranking, then warm engine would have Rich mix so slow to start.
That's all my guess's for now, lol.