As you might have noticed in the process of cutting out fast food and non-water beverages (PS unless you're lactose intolerant I really don't recommend cutting out unflavoured dairy), you don't break a habit, you change a habit. I bet you drink a lot more water now than you did before, amirite?
So, the same applies with snacks. The fact that you eat snacks isn't a problem, but what you eat is. So prepare yourself with an arsenal of healthy snack choices: fruits, nuts and dairy tend to fit well as snacks when vegetables, meat and grains make up your main meals. I personal find fruit very unfilling (due to being mostly sugar and water), so I'll often wash it down with a glass of milk or chew on a handful of nuts with it. I don't eat much cheese or yoghurt, but both of these are fine candidates to add some satiety along with nutrients.