There's a number of possible reasons, but it does depend on what you mean by sounding wet.
If you know another sax player, ask them to play it (using their mouthpiece/reed combo). If it's fine, then you know it's you. Depending on the sax you bought, it may be the saxophone.
The specific reed might be the problem, or you might be playing on reeds that are too soft. A hard reed will make it sound breathy and hard to play, so that's probably not the issue right now.
If there's spit/condensation in the thing, it will sound bubbly or gurgly (if that's a word). Generally sucking back on the mouthpiece will clear that up.
It could also be an embouchure thing. Tone can take a while to develop, and there aren't really any shortcuts.
If you post a recording, we might be able to narrow things down. Otherwise, see if you can find another sax player who can help you out. You'll probably also get told to find a teacher, which isn't the worst idea in the world.
Here's an example of reeds that are too hard and too soft. It's taken from a video from the US Army Field Band called The Saxophone Standard. The whole thing is two hours long, but might be worth a look.