The firmware check (DONE and +) reads: REV 3.63 (US) ENGLISH
...
Backlight timeout after 15 seconds when plugged in to AC outlet.
As it happens I have a console newly purchased, from Scientific Sales within the month. The firmware version reads the same as yours.
However mine kept the console light going for over 30 seconds before I cancelled it to avoid heating. So clearly different from your observation.
I think most likely somehow power is not making it from your wall wart to the true internals of the device. Just where along the path the connection is not made I lack a good reason to guess.
But I do think your behavior is clearly abnormal.
If truly your console is actually running only on battery power, you may be able to observe the reported battery voltage to drop within just a day or two, especially if the batteries are extremely fresh. You can observe the reported console battery voltage at the bottom of the first diagnostic screen (help down TEMP and press HUM to enter, press DONE to exit).
Also, though most of us may never do so, the unit in proper condition will operate on wall wart only, with no batteries installed. If my guess that you are somehow running on battery power alone is correct, then if you remove the batteries altogether, you'd expect no display activity from the console in very short order. If instead it still operates, then you have a somewhat fancier problem, but you still have a problem.
Personally, I have terrible difficulty getting C batteries in and out, and have resorted to a field expedient modification using diagonal cutters to make a little more room for escape near the button end of the battery compartment.