Doubt its transmission fluid: There was a "bad batch" of Axis/Vegas units where the shifter cable was a little bit too close to the exhaust manifold and the outer covering melts. (Its also possible there is/was a holdback that eventually breaks causing this condition as well.)
Your symptoms somewhat match the issue.
We had it happen several years after purchase (I did notice the smell of the coating melting too but didn't think anything about it).
Crawl underneath and look for something like this near the engine/transmission on the drivers side:
It is likely that yours looks worse and that the inner cable (the part that actually does the shifting) has completely separated (which is why you can move the shifter all over the place and it does nothing at the transmission end).
I was able to convince a Ford dealer to fix it under warranty (my guess is that the fix was cheap enough that they warrantied it to shut me up--hey I'll take it LOL).