91 Prizm, 2 problems (idle and tranny) confused..

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I recently picked up a 91' Prizm (or Corolla) for $20. I drove it on the test drive for literally 30 minutes and had no problems with it. It drove great, no noises, no problems, and I was sold. I brought it back home, washed it off, and immediately 2 problems have arisen that I can't solve for the life of me.

I've been working on cars for a while, but I'm stumped on these two issues.

First, idle has been erratic since the car wash. It was fine before, and I did spray the engine down. The idle is really weird. It's perfect when warming up, but right when it hits operating temperature, it goes up and down between 400 and 600 rpms. It's a very quick up and down too. However, I have had no problems with power, stalling, CEL's, etc. Just the up and down idle. I thought for sure replacing all of the vacuum hoses would do it (some were cracked), but it didn't do anything. I seafoamed the engine twice afterwards (maybe there was gunk on the valves), and it indeed runs smoother, but still erratically. The only thing I can venture a guess on is the coolant temp sensor. The wire shielding is burnt, but the wire seems okay.

Second, the transmission is making some funny sounds. It buzzes (hums) at 2600 rpms regardless of the gear and how much throttle I put on it. It buzzes in neutral too. If I keep it at 2600, it will buzz very loudly. If it's at 2700, no buzzing whatsoever. 2500, same thing. There is a lighter buzz at 1500 rpms as well, but very mild. There is also another mild buzz at 3100 rpms ish, but again, much quieter. Sometimes however, this car makes no buzzing sounds at all. Sometimes it's very quiet, but it can still be heard. But most of the time it's very loud. However, it shifts perfectly fine when cold. When it's warm, the shifts are a little stiffer, but it's consistently shifting. The fluid is normal level, and the color is slightly dark. Grayish red, not a burnt color, but old. Because the buzz is so pronounced, I doubt that changing the fluid would remedy it, but I can't be sure.

If anyone has any idea with these problems, please help me out. I am confused beyond reason here.