Ford starter motor has the Solenoid on it, and in your year a Start Relay on the fender.
Some call the starter relay on the fender the starter solenoid, which is fine but not completely clear.
If the starter relay on the fender is vibrating then it needs to be replaced the sliding post inside is stuck.
Make sure truck is in NEUTRAL!!!!!, Emergency brake set!!!
Then use a jumper cable from larger post on starter relay to the other larger post on starter relay, starter motor should crank engine.
If it doesn't then either cables are bad or starter motor is.
"New" doesn't mean "it works" any more, it now means "never tested"
If the vibration is coming from the solenoid on the starter motor then it is either stuck or starter gear is jammed, either way starter will have to come out for testing.
Manuals and automatics often used different starter motors, different depth nose cones and gears, so are not interchangeable, maybe parts store gave you the wrong one.
A relay connects and disconnects an electrical circuit
A solenoid causes a mechanical movement
Both use the same electro-magnetic mechanism, which is why the Ford starter relay is often called the starter solenoid.
Starter relay passes battery voltage to starter motor
Starter solenoid pushes out the starter motor's gear to make contact with the ring gear/flywheel.
In the 1990's Ford moved the starter relay to the starter motor, and Positive battery cable ran right to the starter motor, no relay on the fender.