I can’t really say why things were different between the two headphone outputs, because they are driven by the same amp and the two headphone jacks are simply in parallel.
Look at this snippet from the block diagram...the little triangles are amplifier stages, okay? In this case they are all line amplifier stages, except for the headphone amp:
You see on the right the two headphone jacks, they have the same source, the same pair of amps for the L and R headphone channels. And if you follow it upstream you can see the headphone amp input is the same as the amp for the MONITOR OUT jacks. And furthermore when the headphones and MONITOR OUT jacks are sourced to the STEREO buss, guess what? It’s the exact same signal you should have at the balanced and unbalanced STEREO OUT jacks. So that’s the question...aside from this oddity where the two pairs of headphones were behaving differently, are you experiencing the same “everything is mono” problem at the STEREO OUT jacks as well as the MONITOR OUT jacks and the headphone jacks (at least most of the time)?
Those all source the stereo fader booster amp and right upstream from that is the MONO switch. So is that the case (same problem present at all those outputs)?
If so, I’m wondering if your MONO switch has gone hinky. Have you tried exercising the switch? Like rapidly latching and unlatching the switch like maybe a 100 times?
Don’t get too discouraged yet. Even if you feel this is outside of your wheelhouse there are a couple relatively non-technical things you can try and maybe get some direction.
One of them is exercising the MONO switch as per above. The other thing you can try is tip the 388 on its side, make sure it is well supported, remove the bottom panel, put some audio through to the stereo buss and monitor it with your headphones. Now gently start applying some pressure to the buss PCB, the long skinny PCB that runs across all the input and master section cards. Gently push on it, maybe twist it a little with your thumb and fingers around the monitor PCB (the one closest to the tape transport)...gently push on any connectors on the monitor PCB. Apply some pressure this way and that to the monitor PCB around where it connects to the buss PCB. Have your audio playing the whole time and listen in the headphones. Does the signal change? Does it go from mono to stereo or vice-versa? Is there any crackling, static or intermittency in the audio when you push on any certain spot or in a certain way? Just make note of any and all you notice, what happened and what you were doing when it happened. You don’t have to know the names of anything, but take a picture that kind of shows what you were doing when you were able to make something good or bad happen. Make sense?
Doing this kind of thing can sometimes help isolate a cold solder joint, broken trace or the like.