This is not an opinion everyone shares but I would kick in that tube changes make *less* of a difference on the Mark III than any other amp I've rolled a lot of tubes through (Roadster, RA, JCM800, Ampeg V4).
Not saying it makes no difference, it does, just not much. I definitely would not bother buying one or more NOS tubes only for a Mark III. Rectos are night and day with the right tubes, but honestly I've never heard anything out of tube rolling on a III that couldn't be accomplished with:
1) a nudge of the EQ! If you are looking for 'dark and thick', don't lock yourself in with a tube change, roll down 2200 just a tiny bit, or roll down 80 some and keep Deep pulled at gig volume - normally with the Master much past 2, Deep gets boomy. Also, for heavy riffing, try using Rhythm 2 with the Volume cranked to 10 and a mild to moderate "M" in the EQ for a very thick, aggressive, dark sound that's still tight; or
2) a flick of the tone knob! Probably because it's from the end of the Old Shreddozoic period when pickups were almost all passive and people even played single-coils turned up loud, it's much more responsive than a lot of amps to volume and tone dynamics off the guitar. Remember that basically every Mesa up to the Atlantic series interprets treble early in the signal path as gain, so you can, kind of couterintuitively, get a better defined tone sometimes by rolling *back* on treble.