Yeah, the OHs have a huge effect on the snare sound, and an untreated room will absolutely cause comb filtering and standing waves, producing the "phasey" and "distant" sound you describe. Build yourself some absorbers/traps! It is really critical to making a iffy space usable, more important than any mic or mic preamp, and far cheaper to upgrade. Look here: ht
tp://www.runet.edu/~shelm/acoustics/bass-traps.html
If you're really happy with the drum sound itself, that's a good start. For me, even with a great drummer, in my treated studio, it usually takes a while to get the snare sounding right. I usually spend a large amount of time on it actually, it's often the toughest sound in the whole mix, being so prominent, and well, LOUD. #1 place to improve the sound is on the heads, read the
drum tuning bible if you haven't.
You have to be careful about what you do with one track or overall levels on each, every tweak changes the tone. For that reason, I usually go with simple eq or comp on room or OHs. I usually high-pass overheads so they are mainly cymbals, usually around 200-400Hz, no compression, and pan them about 75 L/R. I also usually low-pass a mono room mic, smash it, and pan it mono. Then I mic top and bottom of the snare, bottom polarity flipped.
I usually start with just OHs and room, get them matched at levels that sound good. Then bring in kick, do some quick tweaks. Then bring in top snare, do some quick tweaks. Get it sounding good without the bottom snare. Them when you bring it in, do just enough so it's only really bringing through the sound of the wires just enough. I usually get in a back-and-forth till it's pretty good on it's own. Then it's on to doing it again with every other instrument, back-and-forth, till everything is right. As you progress, your tweaks should get smaller and smaller, until they don't need anything at all.