I’d check the linkages to be sure that nothing is binding when you shift the transmission but I really don’t think that’s the case here. From what you are saying, it looks like your Neon is exhibiting a classic case of slipping transmission.
What is happening is that the clutch packs — also known as bands — are spinning and not taking up and dropping into gear very quickly. It may be being caused by a number of things including super-low transmission fluid. This could be caused by a plugged filter or be a pinhole leak somewhere. In any case, the transmission is doing its best to move your Neon but it just can’t do it very quickly or well as it takes a long while for the fluid pressure to build to a level to where it can flow effectively, even at a minimal level.
Or it may be a plugged filter or some other damage. Whatever the root cause, the result is still the same, your transmission just sits there doing nothing while the engine revs happily away.
Since you haven’t said whether the tranny fluid is dark or smells burnt I don’t know if there is significant internal damage to the transmission. It is possible that it is just an anomaly and that the transmission fluid is down due to wear. At this point, you have two options. I would suggest a complete replacement of the tranny fluid. You will need a special pressure device to remove all of the fluid from the transmission. If you opt for this try to keep the pressure as low as possible to minimize potential damage.
If the flush and refill works, you will have probably spent about $400, depending on the shop and labor rates. This is still far lower than the $1,300 or so that a partial transmission rebuilt would have cost you. It is possible, though, that the flush may not work and that you will still have to invest the higher figure.