Troubleshooting Transmission Issues - 2002 4.4

[Kirra]

Hey there, I could use an expert opinion to see if I'm on the right track for troubleshooting the automatic transmission issues on my 2002 4.4.

It has 183k with unknown transmission service history. I traded an E46 wagon for it 2 weeks ago.

-Shifts normal in 'D' until it's warmed up.
-Has never given an error code.
-Issues go away (or more likely just masked) when in 'S' or manual. Haven't tested this on long drives though.
-When warm, it'll shudder between 2nd and 3rd, particularly under light throttle and going uphill.
-When warm, it'll hunt in 5th gear. It acts like slipping, but doesn't seem to be affected by throttle.
-When hot, after driving for a very long time, it'll hard downshift into 1st.

My thought process so far...
-It's temperature related, and issues get worse the longer (hotter) it's run.
-I *think* being that the issues are in multiple gears, I could rule out anything related to a single gear.
-The temperature sensor is easily diagnosed (https://xoutpost.com/986442-post13.html)
-If the temperature sensor checks out OK, what is the proper temperature for this trans to run at, and how do I read it while driving?
-Following the last bullet, if I find it's running too hot, are the trans cooler and thermostat the only culprits?

Let me know if you think I'm on the right track, and any assistance you can lend on the questions above. What I don't intend to do is throw parts/time/money at it without proper diagnosis.

Thanks!