If I find the culprit, I will share here.
I had a similar issue with a Tektronix TX1/TX3 or Fluke 183/185 (can't remember which one exactly, but it's somewhere here on the forum) and, after desoldering/resoldering quite a few components, I traced the issue to an SMD ceramic cap which was acting like a resistor.
For my 26M, I think I might have caused this, when accidentally applying about 30V to the battery terminals, instead of the expected 3V.
Interestingly, the meter survived (so, probably, your 3.3V adapter shouldn't be an issue, but don't trust me on that), but something must have suffered during that time.
For the LEDs, I will look to test, but it might take a while until I re-assemble the unit.
In any case, multimeters being able to turn on some LEDs but not measuring the voltage drop is quite a common thing.
Only a few multimeters can provide the voltage (in diode mode) to be able to light up all types of LEDs (particularly the white and blue ones) and measure the voltage drop at the same time.
One such example is the venerable Fluke 87V.