car strats when cold doesn't when hot

I've had the same problem, it was in fact the starter, and it happened with a brand new one.
Not sure the brand you used, but if it's a chain store starter return it.
Ford Motorsport now makes a mini starter and it's pretty cheap, like $150.

One can jump the big lugs on the solenoid (observing proper safety precautions like being in neutral with the e-brake on, etc) to try and rule out the solenoid as being at fault.

Good luck.

My suggestions:

Beg, borrow or steal a voltmeter. Check the bat voltage with the engine off. Fully charged it should be 12 v or better.

Check the bat volts with the engine running, it should be 13.4 or so, and go higher. Should stay up when the headlights are on. If not start looking at the alt and regulator.

If thats ok, check the bat cables, especially the connections. Make sure you have good connections at the battery, and especially at the other end, especially on the grounds!!!!!! Occasionally a cable will pull loose inside the molded clamp, so check and wiggle them hard.

If thats ok, replace the battery. Tell Autozone they can test it after you leave, if they insist on testing.

If that does not fix it, change the starter.

If you are in a hurry, change the battery first. I have found store battery tests tend to say ok most of the time when they should not! The volts may be ok, but it will not put out enough current to crank the car. It is annoying when the store makes a big deal about testing when I am in a hurry to buy a battery.

The above takes about 10 minutes.

After 35 years of owning Fords, this approach fixes 99 of 100 slow start probs.

Rear mounted battery ground wiring. Follow this plan and
you will have zero ground problems.

One 1 gauge or 1/0 gauge wire from battery negative post
to a clean shiny spot on the chassis near the battery. Use
a 5/16” bolt and bolt it down to make the rear ground.
Use a 1 gauge or 1/0 gauge wire from the rear ground
bolt to a clean shiny spot on the block.

One 4 gauge wire from the block where you connected
the battery ground wire to the chassis ground where the
battery was mounted up front. Use a 5/16” bolt and bolt
down the 4 gauge engine to chassis ground, make sure
that it the metal around the bolt is clean & shiny. This
is the alternator power ground.

The computer has a dedicated power ground wire with
a cylindrical quick connect (about 2 ½”long by 1” diameter.
It comes out of the wiring harness near the ignition coil &
starter solenoid (or relay). Be sure to bolt it to the chassis
ground in the same place as you bolted the alternator power
ground. This is an absolute don’t overlook it item for EFI cars

Picture courtesy of timewarped1972




Crimp or even better, solder the lugs on the all the wire.
The local auto stereo shop will have them if the auto parts
store doesn't. Use some heat shrink tubing to cover the lugs
and make things look nice.