The buzzer typically comes 'on' if air pressure drops, so another logical question is "what is your air pressure showing on both dry tanks?". Are any leaks heard?
The trans temp light can come on without the buzzer because my truck has done it. In my case the unfounded transmission overtemp light was due to a loose lower plug on the back of the transmission ECU (inside the dash behind the gear shift buttons). That location applies only to older trucks so you should tell us the year of your truck.
The temp light will go off without having to be reset by cycling power. The transmission is not too hot if you can hold your hand on it for a short time.
Are there transmission codes if you press the up and down buttons together? You will see a series of fault locations, d1 and upward, followed by either 2 pairs of numbers, or two pairs of zeros. Report those and we can look them up.