Echo, echo, echo.
I'm coming in to echo the other comments and lend my own anecdotal evidence & suggestions to the mix.
A) Eliminate & reduce the snails' food source(s). Starve the suckers out. Reduce feeding of the fish so that there's no extra food for the snails and less food that's being metabolized into nutrients that fuel algae growth (which the snails also eat). For awhile, you may try reducing the amount and/or duration of light on the tank. Your goal is to provide enough light for the plants you want in your tank, but as little as possible to reduce algae growth.
B) Trap'em. Catch as many snails as you can by hand. You can lure them with veggies like squash or algae wafers, then when they amass to feed, scoop them out. Or just whenever you walk over to the tank, pull a few snails out. Finally, if you see the eggs, scrape them off the glass, decorations, and plants.
C) Assassin snails, assassin snails, assassin snails. Yes, it's more snails, but holy crap do they work. And the best part is, they don't reproduce as fast as "pest" snails, so it's unlikely that you'd ever have a population explosion. In my tanks, a pair can usually consume on average 1 pest snail per day. Doesn't sound like much, but believe me, before long, you'll have no pest snails. Also, in my experience, they do not harm fry, nor do they harm shrimp (including cherry shrimp). Literally, the only creatures that need to worry are other snails.
