The Taurus essentially has 2 solenoids in its starting system. One is on the fender well under the hood close to the battery. If you turn your key to the start position the solenoid on the fender well will be triggered sending 12VDC to the other solenoid on the starter itself. If the starter is actually spinning without engaging it can be because you have the actual starter motor wired incorrectly.
When you turn the key to the start position the “S” terminal on the fender switch/solenoid is energized… The “I” terminal sends 12vdc down to the solenoid on the starter to trigger it AND throw the gear out into the flywheel.
You must verify that the heavy positive wire from the fender solenoid goes to the side of the starter solenoid that does NOT go directly into the starter motor. This is very important. The heavy wire from the fender must go to the “broken” side of the starter solenoid, the starter motor must only spin after the solenoid on the starter body is triggered.
Also verify that the “I” terminal on the fender switch/solenoid provides the solenoid on the starter body with power so that it can throw the starter gear out into the flywheel.
Methinks you have the heavy lead from the fender hooked to the motor side of the starter solenoid switch…which is why it spins but doesnt engage or “throw” as they say.
I apologize if this is a bit confusing but having two solenoids in the system makes writing a description for advice that much harder. Just know that the fender switch is simply cutting power to the starter solenoid on the starter body, so it is not hot at all times. Its a bit silly. But the only way your starter would spin and not “throw” is if the starter motor solenoid was not wired correctly OR if you have the “S” and “I” wires mixed up on the fender switch.
Ford uses this system so that there is not a “Hot” wire going to the starter at all times… The fender switch/solenoid makes and breaks the feed to the starter motor solenoid on the starter body. After you understand that… the rest is traditional starter motor/solenoid wiring. The only way to spin your starter motor and not throw the gear out is if you are energizing the motor side of the starter solenoid connection…it needs to go to the “broken” side of the starter solenoid on the starter body.
I apologize if my description gets fuzzy because I am using the term starter solenoid all over the place. The fender switch/solenoid should be called the feed solenoid in my opinion…perhaps Im using incorrect terms but I think you get the idea. If you need further assistance I will do my best… I may need to draw a diagram honestly.