Analog gear will always be susceptible to tuning drift to a certain extent. Some equipment are better than others at reducing it, or controlling it. But that's part of "the game" of using analog instruments. Be it a guitar, a piano, a violin, or an electronic analog music synthesizer. Every professional musician tunes his or her instrument before and sometimes even during a performance (or have a tech back-stage doing it for them in the case of guitars). Of course a grand piano doesn't drift as much as a Moog. But it costs over 10 times more than a Voyager!
If you want to go the lazy way, and use extremely precise, digital, cold sounding synths, you're free to do so. But if you want warm, rich sounding, pleasing to the ear gear, you'll have to work harder than just pressing the record button and play notes.
And you're very lucky that modern Moogs are very stable when compared to the early modular systems. Just ask Wendy Carlos. When she recorded the acclaimed and revolutionary Switched-on Bach album, back in 1967, on a multi-track analog tape machine, she had to tune, and retune the whole Moog before every single take! Sometimes, she even had to stop in the middle of a track, and re-record it because the instrument had gone out of tune during the recording of that track! Because, back then, regulated power-supplies weren't very efficient and the voltages inside the modular would fluctuate and change the tuning, on top of temperature drift ...
So you see, just having to touch up the "Fine Tune" knob once in a while isn't so bad, is it ? (Edit: unless your RMEs drift is extreme, see last words below) I have recorded several multi-track songs with my 37 years old Minimoog D, and yes I had to make sure it was in tune before each take using the on-board A-440 reference, but like I said before; it's part of the game...and the charm of using such tools. But I wouldn't want to use anything else when I want "that" sound.
I doubt that exchanging your RME for another one would change things much. Maybe some unit might be less susceptible to temperature drift than others, but I doubt it would be by much.
All that being said, it IS possible that your RME might have developed a fault in the temp compensation circuits. I would talk to a Moog tech to see if it's out of specs, first.
TIFWIW.
Al.