1986 Ford Thunderbird, 5.0 HO. Starter stays on. I have replaced the solenoid and starter. I leave the battery disconnected and when re-connecting, sometimes the starter begins to run, and other times does not. If it does not run when re-connecting, and I start the engine, the starter again becomes stuck "on." It will continue to run with the ignition "off" and the key pulled out.

Try following the positive battery cable from the battery to the starter solenoid (not sure if you have an inner fender-mounted solenoid or a starter-mounted one. Take one lead of a pair of jumper cables and connect one end to the solenoid terminal with the cable from the battery, and take the other end of that same jumper cable lead and momentarily connect it to the other large terminal on the starter solenoid. If the starter engages when you do this, replace the starter solenoid. If the engine doesn't try to crank when you do this, you probably have a bad starter. Good luck.

sounds like your fuel filter might be plugged causing the engine shutdown.
sounds like the solenoid on the starter is sticking causing your starter to stay engaged and powered up. the starter will stay powered up and will continue to spin until the starter solenoid is disengaged.
one other area to look at is the ignition switch. if that is bad it will continue to send a power signal to the starter solenoid keeping it engaged. so, you may want try unplugging that first, while your starter is running. if continues to run, then i would say it is the starter solenoid and not the switch. good luck

The starter is mounted on the driver's side bottom of the engine.

1. Disconnect the negative battery cable.
   
2. Raise the vehicle and support it safely with jackstands.
   

Starter mounting-1986-89 RX-7

1. Disconnect the heavy battery cable from the terminal marked B on the starter solenoid (or magnetic switch on later models).
   
2. Disconnect the thinner ignition switch wire from the terminal marked S on the solenoid/magnetic switch.
   
3. On vehicles with automatic transmissions, remove the front starter motor bracket bolts and the bracket. Remove the two starter attaching bolts and remove the starter.
   

To install:

1. Support the starter by and hand and position it onto the flywheel housing. Install the two mounting bolts. On 1986-89 RX-7s, torque the bolts to 24-33 ft. lbs. (33-45 Nm). Install the front starter bracket, if removed.
   
2. Connect the starter wiring to the appropriate solenoid or switch terminals. On 1986-89 models, torque the B terminal (battery cable) nut to 8 ft. lbs. (11 Nm).
   
3. Lower the vehicle.
   
4. Connect the negative battery cable.
   

The one mounted to the wheel well is the starter relay.

From what the diagram in my Chilton Manual. You can change the solenoid.

Sheilah

Remove starter by 1) removing battery cables from battery. 2) jack up front of car, put safety jacks under car frame. 3) block back wheel and put on emergency brake. 4) remove the plastic cover next to tranny. 5) remove 2 bolts that hold stater to engine, lower starter, and remove positive cable, and starter solenoid wire (the small one) mark everyhting so you know where they go(take a picture with you phone camera) 6) take starter to a reliable auto parts place where they can test it.

Dear Fuel Pump Locator.....
The fuel pump for your tbird is located inside the fuel tank of the vehicle....as well as the sending unit for the fuel .Yes it is a bear to get down on the ground...especially without a car hoist.The starter solenoid should be on the LF inner febder under neath the hood of the vehicle...I HPE I HELPED!!

sounds like the starter is defective, pull it out and have Kragen or autozone test it, it is free

Starter

Removal & Installation 1. Disconnect the positive and negative battery cables.
2. Raise and safely support the vehicle.
3. Remove the starter terminal cover.
4. Remove the terminal nut and separate the battery starter cable from the starter motor.
5. Disconnect the solenoid (S) terminal connector, if equipped with a starter mounted solenoid.
ø To disconnect the hard-shell connector from the solenoid (S) terminal, grasp the plastic shell and pull off; do not pull on the wire. Pull straight off to prevent damage to the connector and (S) terminal.
6. Remove the starter motor retaining bolts.
7. Remove the starter motor from the vehicle.
To Install:
8. Place the starter motor in position and install the starter motor retaining bolts. Tighten the bolts to 15-20 ft. lbs. (20-27 Nm).
9. Install the battery starter cable and a terminal nut to the starter motor. Tighten the terminal nuts to 79 inch lbs. (9 Nm).
10. Connect the solenoid (S) terminal connector, if equipped with a starter mounted solenoid.
11. Install the starter solenoid safety cap. If equipped.
12. Lower the vehicle.
13. Connect the positive and negative battery cables.
14. Start the engine several times to check starter motor operation

if the car is an auto please check the switch on your auto box also starter soleniod part of starter motor try and check power to soleniod when turning key and check your soleniod connections

have already answered and advised to your earlier post re correct connections to solenoid on starter