Snare resonator heads are supposed to be ridiculously thin. Don't worry about that.
The best way to control the sustain of any drum is with tuning (it's also important for best tone). Snares also have, well, snares that can be tuned a bit. But definitely start with tuning the drum.
Basically, every drum has a lowest resonant frequency based on its diameter and depth, and there is also an overtone series above that frequency where the drum will resonate. A good way to find these is to detune both heads, throw of the snares and dampen the resonator head, and then slowly tune up the batter head until it really takes off when you hit it.
If you like that frequency, then good, you're done with phase one. If you want it higher, keep tightening until you hit the next resonant frequency. If you want it lower, too bad, you need to buy a different drum.
So by now you have the batter head tuned to a resonant frequency. Flip the drum over, mute the batter head, mute the snares (really just loosen the screws on one side and pull the snares away), and tune up the resonator head until it's at the same frequency as the batter head. Now when you hit the drum, it really takes off and sounds big and lasts a little while.
Ok, maybe you're thinking "Todd, you fool, I want a short sustain!" To which I smugly reply with the fact that the shortest sustain tuning is very close to the longest sustain. You just want to detune one of the heads a bit until it really shortens the sustain. In the case of a snare, I would probably detune by tightening the batter head a little bit.
So now you have what is effectively a high frequency tom with a very short sustain. Clamp down the snares again and then start tightening them up. Many snare levers also have a thumbscrew like thing that tightens and loosens the snares. To get them as tight as possible, you might have to throw them off, tighten a bit, and throw them back on and see how it sounds. Tighter is shorter.
If the snares are still too long in sustain, then you can buy different snares. You would want thinner and fewer snare wires on a replacement snare set.
Damping drum heads is a quick fix but can really hurt the tone and projection of the drum. If you tune routinely and carefully and frequently, your drums will have a more open sound with more projection and character, while still having the sustain and qualities you want for situation.
And finally, I can't talk about drum tuning without recommending a drum dial like device that measures head tension. Not only does it help make sure the tuning is even across the whole head, but once you have a head tuned perfectly, you can write down the tension and easily and quickly tune up to it before gigs, etc.