Your car is going into limp-home mode very quickly because there’s something very wrong with its transmission. Normally, when a car goes into limp-home mode it has experienced some event where the transmission control module (TCM, the computer that controls your transmission and works with the engine control (ECC) module) takes various inputs from the sensors it monitors (mass air flow, mass air pressure, speed tensor, throttle positioning sensor and such) as well as from the sensors the ECC monitors that work with the transmission decides there’s a problem and puts your transmission into a state where it will limp along in a forward gear or in rear and a forward gear, giving you the ability to get to a service area or to your home where you can have the vehicle towed in.
What happens is that a computer code is sent to the ECC or TCM that says the “XYZ is bad, go into limp-home.” Then, your vehicle performs very much as you indicate. It can be more than just a simple code, though. In fact — I think this is the case with your car — it may be that a mechanical part has plainly gone and the resulting performance is because of the bad system.
In your vehicle, I think it is a combination of systems, so many systems that you can pretty much think rebuilt computer and $3,000 to $3,800 expense from the start. As to the specific systems that may be involved, it sounds like there could be a problem with the torque converter and the input turbine impellers that, combined with a problem with the clutches/bands and gears, is rendering your car undrivable.
I could sugar-coat it and tell you to have this system checked and then that system checked to keep your costs down, but, when you add up all of the potential diagnostic work that might be involved, you are looking at a lot more than the $3,400 average figure. The cost is greater because you have to add in the various checks to the cost of the replacement transmission you will be having installed.