My car goes into a limp mode where it will stay.

It stays in second gear when the computer detects a problem and sets a diagnostic fault code in memory. Once the computer was disconnected those codes were erased and that valuable information was lost. The same code(s) should be stored in the new computer. Having them read is the place to start.

The computer should be the last thing to suspect. They cause less than one percent of the problems so it's expected that it didn't solve yours. The most common failures are with the two speed sensors. About half the time it's a problem in the transmission itself and it has to be rebuilt. This is not conclusive but as a general rule if it starts out in second gear as soon as you shift into drive, it is a sensor or other electrical problem. You can tell it's in second by how sluggishly it starts moving. Typically the speedometer won't work either. If it starts out in first gear like normal, then bangs back into second gear during or right after an up-shift, that is usually caused by excessive slippage in one of the clutch packs. Your mechanic can get an idea of the amount of wear by viewing the "clutch volume index" (CVI). That is a set of four numbers that correspond to the volume of fluid it takes to apply each clutch pack. An experienced mechanic can tell how much life you can expect is remaining.

That should be good news. A problem inside the transmission shouldn't care about temperature. I am inclined to suspect a sensor problem but you need to have the fault code(s) read. Those will get you into the circuit or system that needs further diagnosis.