Transmission issue

First thing would be to check the fluid level and condition. If it is low or burnt it can act this way. The transmission fluid hot could indicate both issues as lower fluid levels would heat up faster.
The next item would be that you have an electrical issue inside the transmission with either a sticking pressure solenoid or mechanical damage causing the heating.
That can be tested using an OEM level scan tool to watch what the transmission is being told to do by the computer versus what it is actually doing. Not something that really falls into DIY territory. Either a good independent shop or transmission shop would be the next stop. They can scan it and give you a better idea of what is actually wrong.

It is possible that the issue is only a faulty temperature sensor shutting things down to protect itself, but without the scan tool you cannot tell.

Transmissions definitely are not in my area of expertise, but I do know that this can happen if the pump is not able to build enough pressure. In this case, it sounds like the screen in the transmission pump gets clogged up when the car runs, and then cannot put out enough pressure to drive the clutches and gears like it is supposed to. There is a small chance that getting the transmission serviced could get it going better. But honestly, I think it is probably been toasted by now due to overheating the transmission.

You can also just try checking the fluid level. If it is low, cap it off and see what happens. Start the car, put it in 'D' for a few seconds, then in 'R', and then in 'D' again for a few more seconds. Then put it in park. Raise the hood, locate your transmission fluid dipstick (somewhere on the driver's side of the car, should have a red colored end). Pull the dipstick, clean it off, and put it back in. Pull it out again and read the fluid level. Go from there.

If you have to buy transmission fluid, I suggest going into a parts store and getting someone to look up whatever fluid your car needs, and get that. There are several different fluids for different cars and using the wrong one can cause serious problems.

Good luck and let us know what you find.

Black is not good. That is the sign of burnt fluid and clutch/band failure, especially if it feels gritty when rubbed between your fingers. Either a rebuild or replacement if you like the car. Last one I had done was around $1,700.00 for a full rebuild with a new converter. Be sure if you have it done that the cooler gets flushed and it is not a bad idea to add a second one after the OEM. Heat is the number one destroyer of automatics.