This is an opinion on an anonymous sugar pastry forum, with a basic description of the problem and cause and no direct interaction with the vehicle.
I think your torque converter is fucked. It can cost UP TO 3 grand to fix it judging by what i have seen over the years. Btw i'm in Canada so it would be less for you i think.
It could also be: but i don't think it is, simply because the engine is revving and the car is moving, just slower than it should be. This in my mind implies that fluid is not forming a proper seal and is leaking somewhere between the impeller turbine and stator, to make it move at the correct rate, however since there is still some fluid being sealed the car is moving at least a little bit.
The control unit being bust would mean the car would be shifting badly, or not at all, but it's in gear just not moving the car at the correct speed.
Take it to a professional, just be ready for a hefty bill. And try not to believe everything you read.
If you ever hear X saves fuel, think of the reasoning behind it, in this case the idea is that since the tranny is not in gear the friction isn't there. This is simply dismissed because a good transmission doesn't have that much friction anyway, and no matter in gear or not parts in the transmission are being driven. Also research the parts involved in a process like this and think if you can damage them by doing whatever.
If you can't find some proof of something working, just don't do it.
If you wanna save fuel. Accelerate, keep it below 3000, gently let off the gas and the auto tranny will drop a gear. Again do it to maximum 4000, and then let off gas tranny will drop a gear, now in third unless it's over drive get to the speed limit (lol) and let off gas until it's in over drive. If it's a 3 speed with over drive, get to second, use second to get to desired speed and let it drop to 3rd.
We do this manuals, it's called short shifting. It keeps the Revs low and thus saves gas.