If your snail populations are exploding, chances are they have too much to eat. Keep an eye on your feeding levels and cleaning regimen.
MTS are hard to get rid of. You can trap and remove them but you will have to do it forever and it will only keep the levels in check.
Assassin snails will eat MTS and are a decent community member as long as you don't have any snails you want to keep.
Another solution is to break down the tank and soak everything in a bleach solution. That might be a more nuclear approach than you were looking for though.