I recently purchased a Mohawk for AM use and have noticed that when switching between LSB/USB while in the AM mode, I have to retune to peak the signal.
It depends on how you are tuning the signal. If you tune an AM signal in for zero beat with the BFO ON (and the BFO is centered properly) -- then you will be tuning to select the sideband you have set on the selector switch. Turn off the BFO for normal AM listening. You can then change sidebands with the USB/LSB switch and get the other sideband. How well this works depends on a number of factors. Later Hallicrafters radios did the same "selectable sideband" thing on AM, and it doesn't really work perfectly, IMO, and is mostly worthless.
I'll guess that what you are doing is tuning in the signal like you would any AM signal -- BFO off, tune until the signal peaks and sounds "tuned in". By tuning that way, you are centering the AM signal (it's carrier) in the passband. It doesn't really matter which sideband you have the switch set to, you are getting as much of the AM signal, BOTH sidebands, that will fit in the passband. So if you've tuned this way, by virtue of the way the IF frequency conversion circuits work, when you switch to the other sideband, you will have to retune to "center" the signal again.
So I suggest just pick a sideband and leave the switch there. Tune in the AM signal. You can then adjust the TUNING, up or down to shift the signal in the passband and get one (or mostly one) sideband. Shifting this way can help eliminate interference if any, and also by getting mostly one sideband in the passband you get maximum fidelity. It's just less complicated than trying to tune the signal perfectly (and on the correct side of the carrier) so that you can use the mode switch to change sidebands directly on AM.
As to the difference in signal levels when you switch sidebands, I'd suggest swapping out the 12AT7 1632-1732 kHz crystal oscillator tube as a first step. It could be that one half the tube is weak, and that the injection levels from one of the two oscillators is significantly lower than the other. The only thing that actually changes when you change sidebands is which half of that tube (and thus which of the two oscillators) is running. Other things could affect it -- a low activity crystal, component value shifts, even dirty switch contact, etc., but the tube is a good first start if you don't have any test gear to check voltages, etc.
Grant NQ5T