Thank you so much for the reply, Deft Spade!
I've managed to mostly solve my issue by choosing louder samples and cranking up their volume, to drown out the noise. (If you make your song in beepbox, the instruments that work well for this include, at the very least: contrabass, voice synth, french horn, cello, and standard drumset; just export as .mid and convert to xm by openmpt and it should work fine.) And I'm also learning to embrace the noise. It is perhaps part of the "charm" of developing for these old systems... Oh, and maybe I'll try increasing the mixing rate. Thanks so much for the help, Spade, and everyone else! Making music this way is pretty fun :)