I have a 2001 Lincoln Ls 3.0 . The car will not move while trying to use the automatic gears but will move while using the manual gears but has no reverse . This started after the car was accelerated quickly and eased down as I came to a stop sign . The car would not take off from there . What could be the cause of this ? Any solutions ? Do I need a new transmission ?

SOURCE: Continuous "whine" @ 1500 RPM only; and jerks/2001 Lincoln LS

Run the car from the top of a hill and let it coast at your whining speed. Put it in Neutral, does the whine disappear?
If not its in the drivetrain (wheels, driveshafts, tranny)
Very unlikely to be the timing belt. If it appears at a fixed speed then its drivetrain, if it whines at fixed revs its the engine.
Most likey the transmission I fear....Tires will also make loud whining noise too though...

SOURCE: lincoln ls shifts hard into reverse

You will have to have the solenoid body replaced in the tranny and have the PCM reprogrammed.

SOURCE: transmission, automatic. 2001 Lincoln LS 6 cyl.

check the speed sensore inner and outter

SOURCE: 2001 LINCOLN LS V6 - ROUGH IDLE / SOMETIMES STALLING.

Very common problem is about a dime size hole wearing in the rubber elbow on the PCV valve located on the driver side rear of the intake manifold.

SOURCE: putting car into reverse after having driven short

You need a new Pressure Control Solenoid Body. 9L2Z7G391A -solenoid pack for a 5R55S Trans. $332.20. at Trans Parts supplier. Don't even call a Lincoln Dealer for a price. They wouldn't even deal with me because I have a sales tax ID, and ask too many question. They will screw you over big time. That's a common problem in a long list of LS issues, my wife has one also and I used to have a Trans Repair Shop. It will slam into Reverse cold, Slip or flair on 3 to 4 Shift. Also expect the EGR Valve to go bad and make the trans issue even harder to troubleshoot, untill it works correctly. Drop the tran pan, remove about 7 screws and change the solenoid body. Best bet, go to a trans shop, never a dealer. Also get the old fluid out of there and put Mecon V in ,with a bottle of Red Lubgard Additive.