2003 ford taurus starts, won't move

Question edited for clarity.
Question moved to model category.

It is a known issue

https://www.google.com/search?q=1998+Ford+Taurus+Wagon.+Cannot+move+column+gear+shift+from+Park

do you feel the shifter moving anything? Possible the bolt came undone at the top of the transmission.

Just to the left of the battery is the main fuse junction box and below that under the manifold is the starter with the solenoid on top with the positive cable going to the starter check for a loose connection, and the battery ground sould be right there nearby the starter grounded to the block check it also. You could have a bad cable although it may look okay under the insulator the copper cable may be damaged test with a volt meter while moving the pos and neg cables at both ends.

Well, yes the solenoids would cause the transaxle to not function right.
Are you sure the trans is not engaging, or are the brakes locked on ?
You say it shifts fine, you mean you can move the shift lever ?

Your transmission is toast. Smell the fluid on your ****** dipstick. It will smell really burnt.

You say transmission fluid levels are good. Is it a nice red or pink color or is it an orange, brown to black color. Does the vehicle actually shift into gear, the linkage from the gear shift from park to drive or reverse maybe be unhooked you can check that out also. Does the trans fluid smell burnt, should be kind of a sweet smell. Good Luck

check speed senor and vss

the starter has nothing to do with it... taurus' are known for stripped out torque converters. most likely the case. about $1000 job

The DTR switch on the side of the transmission, that the shifter hooks to, to change the gear positions may have gone bad. Usually about fifty dollars,but you have to get it adjusted right,and that means using a shop manual to install it. don-ohio

it sounds like the alt drive it too auto zone and they will test it for you or you can take it off and have it tested also