Shifting transmission to neutral or drive drives forward, but reverse does not move, only revs engine?

This happened 2 days ago. First, I was driving out of the driveway and my car didn't want to move, however the reverse worked fine. After ~5 minutes of lightly revving engine, car started to drive forward but could smell burning odor. Drove car approximately 10 mins then tried reversing into driveway but reverse would not work. Had to drive forward into the driveway.

Since then, I am currently stuck as I can't reverse out. As per title, car wants to drive forward in neutral and drive gears but will not reverse, only revs engine. Checked transmission fluid and when vehicle is cold and off, shows normal reading. When hot and running, shows empty. Added 1 quart of fluid but still reads the same.

I will add, I'm unsure of the last time the fluid was changed. I've had the car for nearly 80,000 miles and to my knowledge it's never been changed in that time. When I checked the transmission fluid it has a burnt smell but is still a normal red color, maybe a little dirty but not significantly. I topped up with 1 quart of Castrol ATF all vehicle fluid, certified for Toyota vehicles.

Any help is greatly appreciated as I'm stuck out of town on a long weekend and all shops in area are closed.