You can try a couple drain and fills, and see if the condition gets better.
there are DIY's on this site on how to do it.

If it doesn't help you'd be looking at additional repair to an rebuild, especially since your close to 200K.

 

I can't speak to the specifics, but in most automatics, the line pressure to R components is higher than that to the D components.
If you increase rpms to ~1500, the line pressure should increase, subject to the control of the computer which regulates line pressure. You could have a bad sensor, a bad pump, or a leaky clutch pack or servo(a piston & cylinder with an arm that clamps onto the end to squeeze a band). ( I don't remember what is 'engaged' in R, and don't want to look it up).
It could be that a solenoid valve is gummy or has some 'stuff' that plugs it a bit, but I don't know the odds.
Most times when a servo 'slams', it is because there was insufficient pressure to overcome the spring that moves the piston back to 'rest' until the pump got going enough.
The above surely didn't answer your question, which I summarize as: Is it broke enough to need rebuilding, or can I swap 'this or that' and have it operate better?
The only things you can swap from outside, as I remember, is the valve body & solenoids, and the speed sensor.
Check your fluid level when the transmission is warmed up completely. It will read overfull when cool, engine stopped, so the engine must be running also.
tom

 

Thanks for the thorough description, Tomw! I guess low line pressure could also explain why it functions fine when the transmission fluid is cold and slightly thicker, vs when it is hot and thins out a bit.