Yeah, I would start using a smaller cup and/or diluting the milk with water. And, if she's asking for it all the time, I think you might just have to say no sometimes -- I mean, say, "this time I'll get you water and you can have some milk later" or whatever feels right to you.
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My kids are often hungry while I'm cooking meals, and I try to resist the urge to give them a carb (chips, crackers, bread -- so easy) and instead offer them raw vegetables or fruit. I will either just get a few things out and put them on the table, or offer them a choice between three things. Dd is 4.5 and she will resist sometimes, but I just tell her that those are her choices if she's hungry.
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At meals I give my kids a small portion of everything, and if they want more of the carb or meat, I will tell them they have to eat something else first (their choice of which thing). That way they aren't eating a ridiculous amount of pasta or turkey hotdogs without anything else. I figure if my dd really wants to stop eating after one hotdog to avoid vegetables, that's her choice.
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I also try to make sure I prepare at least vegetables (or sometimes fruits) with every meal, and often with snacks too. I don't put dishes of food on the table because 2yo ds will just grab everything and make a scene, so I will do a snack plate for each of them like, 2 pieces of cucumber, 2 slices of peach, 2 small slices of cheese, and 2 crackers. I encourage them to eat something else before having more crackers or cheese, or I tell them I will get it after I finish mine -- putting it off will sometimes lead to them eating what's left on their plate.
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I also try to put water out for them regularly. We made cider and they ask for juice quite a bit, especially 2yo ds. They also like almond milk by the glass, but I really prefer they drink water. I dilute their juice when I give it to them, and I try to limit it as much as I can. I just choose not to have it around most of the time to avoid the issue. Actually, you could do that with milk. You can put it toward the back of the fridge so it isn't the first thing she sees when she opens it, and you could actually let yourself run out for a couple days from time to time. It might help make the switch.