I don't believe the temp indicator is related to your transmission issue. I think the temp indicator has its own sensor and there is another sensor for sending info to the computer.

But the computer does nothing to control the trans in that car. The trans is supposed to delay the first 2-3 shift of the day to about 2500RPM in an effort to speed the warming of the cat. Assuming you have an American-market car with a cat.

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2012 E350
2006 Callaway SC560