I'll keep my fingers crossed for you, but a less-than-ideal snare sound might just be the nature of the beast for a drum at that price point.
With that aside and trying to help you tweak this thing... Put the drum on a stand, get your ear below the drum, and tap the top head. Try to determine if the buzzing is from the snare against the head or if it's a mechanical buzz from the snare mechanism or other hardware. If the sound is coming from the snares against the head, you can try taping across the snare to reduce snare response. Some amount of buzzing is pretty normal for wire snares, though. They're notorious for "sympathetic buzzing" when the bottom head is excited by other sounds.
If you really want a better sound and have the option to return the drum, I'd strongly consider it. You don't necessarily need a modern high-tension drum to handle the kind of heads we've been talking about, but a higher-quality drum will offer gut snares with a whole lot more adjustability. (A higher quality shell, 12 tuning lugs, and cast rims would be nice, too.)
To keep the price down, a used drum could be a bargain. I wouldn't be surprised if you could pick up a nice old Ludwig Challenger, Yamaha MS-8014, pre-FFX Pearl Championship, or Slingerland TDR for around a hundred bucks or less. There's slim pickin's on eBay right now, but these kinds of drums show up all the time and there's not a huge demand for them. (The drums that sell for lots of money seem to be modern free-floaters or pre-1970's drums in good cosmetic shape.)
If you don't want to wait for a bargain on eBay,
HERE is a Premier HTS 484 with harness for $200. It's a high-tension drum, but Mylar can sing on these things, too. (Star of Indiana used Mylar heads on Premier free-floaters.)
Taylor Music has a few 14" and 15" Ludwigs listed in the low $100's and some Yamahas in the mid-$100's, including a Corps Custom MS-9014 (my favorite non-free-floating snare drum ever). They don't have any pics posted, though.