Coated Emporer with Moon gell (Like Catron said

) or coated PowerStroke 3 (a la Steve Gadd/Sonny Emory).
*** I understand wher Catron is coming from suggesting the non-coated heads and for toms that is applicable BUT that limits the versatily and musicality of the snare drum which is the main voice of the drum kit... I don't recommend you go that route.
Remos as a whole are too bright for me becaus eof the stiff mylar. Warmer sounds come from more plyable heads (EVANS, ATTACK, AQUARIAN).
The ring you aer getting is similar to the ring that comes out of the Omar Hakim 13"x5" Power Picollo. Does it go away.... hmm NO, But they can be helped. African Mahogany is a very dense hardwood. It vibrates at low frequencies but it is really reflective. So you are going to have to tune lower, especially the top head. Turn back the bottom a full turn. Try 1.5 to 1.75 turns after finger tight on top and add a block of moon gel an inch from the edge with the heads you have. If you you don't have moon gel fold one square of toilet tissue into a square the size of a quarter and use electrical tape.
I try not to muffle my snares at all. I really don't like to muffle any of them but sometimes you have to.
EVANS, EVANS, EVANS,
Head suggestion.... EVANS Genera/Hazy 300 medium tension tuning a la Billy Ward. Or Genera DRY (vented)/Hazy 300 . These single-ply combinations tend to speak very clearly without over muffling the drum the get rid of overtones.
The other option would G2 coated normally, It worked in calming my Oak snare. HOWEVER... because of the ring that you have the Genera HD/Hazy 300. The Genera HD is a 2-ply coated with an internal ring like the single-ply Genera. This should keep the warmth and get rid og the ringy overtones.
Note: because of the wood's density your rim shots will have a bite to them comaparable to a metal drum.