Q: Engine "Locks" Periodically

Hello, I have a 1994 Ford F-150, 5.0L-V8, fuel injected, auto trans. Sometimes (not every time) I go to turn it on and the engine will not turn over. I hear a click sound coming from the starter solenoid attached to the starter motor. I have already checked all the basic stuff, see below.

1. Battery: headlights, radio, wipers all work with key on. AAA was unable to jump the vehicle. I replaced the battery with a brand new, more heavy duty battery, same result.
2. Starter: autozone checked the starter motor and solenoid for functionality, they both work. I tried replacing the other solenoid inside the engine bay and that did not work either.

Essentially, what is happening is that the engine "locks up". When the starter solenoid engages, the flywheel doesn't turn. However, the engine is not seized. Later today I will attempt to turn the crankshaft by hand. Any ideas? Put penetrating oil in the cylinders? Oil change?

Thank you!

My car has 210000 miles.
My car has an automatic transmission.