New starter & Solenoid - Still hanging up

When this occurs, does the starter side of the solenoid show 12 volts (even after the key is released to the run position)?

If it does, remove the little wire on top of the solenoid and see if the issue abates. If it doesnt, there's a chance the solenoid is latched.

Good luck and welcome to the boards.

is your ignition switch sticking?

The common and Normally open contacts in the starter solenoid can stick together (latch). This makes the starter run until you disconnect the battery. If you were to disconnect all wires to the solenoid and put a meter across the two terminals, there should NOT be continuity. (I dont necessarily recommend testing this, as it could create more variables than it's worth).

Since it looks like disconnecting the small wire atop the solenoid did NOT help the situation, that removes the ignition as the culprit. A latched relay is where I'd focus (and ensuring your wiring is correct, if there is doubt).

The trigger wire goes on the top post on the solenoid (it's a small wire with a little bootie IIRC). Then the starter cable goes all by itself on the front solenoid (big) post. All the other crap goes on the rear (battery powered) post. If you have an aftermarket or mini starter, this configuration changes slightly (I assume your 87 to be stock in terms of the starter wiring).

Good luck.