I drive an '84 T-Bird which I’ve mentioned several times on the SDMB.
About 10 days ago, I was on the way home–in fact I was approaching the corner to turn onto Western Avenue and then turn into my mobile-home park about a half-block away.
There was heavy traffic on Artesia before the turn onto Western Avenue. I had to stop, in the heavy traffic, but when I accelerated to move forward again the car just sat there, engine still running… I tried moving the lever into all transmission positions–still no movement. I called the Auto Club on my cell phone; the guy came and he couldn’t do anything with it either.
I got the car maneuvered into the back of our driveway–we’re very close to that entrance, thankfully. But the lever won’t shift the transmission out of Park; in fact, the lever feels loose in any position other than park. (Fortunately, the tow truck hauled the car to our driveway with the [unmoving] drive wheels up.)
With an 84 T-Bird, is this the transmission or something gone awry with the lever?
With the lever feeling loose, it’s certainly possible that it or some other part of the shift linkage has come adrift or broken. This is not an uncommon problem.
Reading your post, it sounds like you were in gear and then it suddenly and unexplicably went into neutral, or acted like it was in neutral. I don’t see how a shift linkage problem would cause this (though I won’t say it’s impossible).
What doesn’t make any sense to me is that you shifted into all positions, - including into and out of Park, I presume - then rolled it into your drive, which means it couldn’t have been in Park, and then apparently got it into Park but now can’t get it out of Park.
Probably not related, but I once had the same condition occur. Shift into any gear, no movement. Turned out to be an axel joint (the turnbuckle in the middle of the axel on the right side of the transmission). This was in a 94 Chevy Cavalier.
Sounds like time to call <beep beep> double-A M-C-O (AAMCO).
The first truck that came was a flatbed and it hauled the car onto its platform with the rear wheels up. Because of the width, or lack of same, of the driveway between our carport and that of the mobile home opposite, the flatbed couldn’t push the car to the back of the driveway–we have another car, which I wanted enabled to be in front of the T-Bird. That tow driver left the car in visitors’ parking–which the park management frowns on.
The second truck they sent was more conventional and it got the car to the front of the driveway; we had to leave the other car in visitors’ parking until the following day, when they sent a third truck out. Another flatbed! :rolleyes:
This driver, however, had a better idea: He brought two carriages out of a side compartment in the truck and put one under each (stationary) rear wheel. He and I were able to push the car to the back of the driveway, where it still sits.
At long last, the mystery is solved. A neighbor brought a mechanic over; the guy took one look under the car, at the point where the shift linkage goes into the transmission housing, and found that a fitting on the catalytic converter had slipped and was interfering with the linkage. Eureka! 
The car drives again.