F 150 starts and starter stays engaged unplug alternator starts ok but no charge

Did you have the charging system diagnosed or did you just guess at it being the alternator . The charging system is computer controlled . The starter isn't working because low power , battery dead . Is there B+ voltage at the wire on the back of the alternator .B+ voltage at pin 3 at the connector pluged into the alternator .
Charge the battery use a volt meter an test for B+ voltage at the places i mention . No check engine light ? DTC'S - diagnostic trouble codes stored .
Ford Smart Charge System Alternator Not Charging
Ford PCM controlled alternator testing

starter bendix is stuck need a new starter or rebuild starter with new bendix

the clicking noise is the starter relay engaging, you say it starts w/jumpercables-starter is ok, either alt is not charging the battery or the battery can not hold a charge anymore any auto parts store can check it and tell you witch is bad

If you can jump start your Lincoln...the battery must be low...(when running have it tested for free at your local auto parts store...the alternator controls the full charging of your battery when the motor is running...If you can not successfully jump start your Lincoln it points to definitely a defective battery, and the starter could be bad also...if your battery is fully charged, and the starter will not turn the motor over the battery connections or the starter is bad...Not a big swithch out...Hope this helps.

will it start with a jump? check battery for full charge, maybe alternator is not charging, check battery connections all the way down to starter, check selonoid by cross jumping terminals to engage starter.

solenoid may be sticking. Have someone start the car, if it sticks tap the solenoid with screw driver handle. If that stops the starter, replace the solenoid. If it were me, I wouldn't bother testing it and just replace solenoid.

sounds like a shorted field in the alternator

shim the starter it sounds like it is staying engaged to the flywheel try shiming the starter

Check the starter selonoid