For your weight, 1000 calories/day isn't enough and you probably aren't eating enough proteins and fiber to keep you satisfied, which leads to binging.
Make sure you eat protein with every meal/snack, because protein keeps you fuller longer and it also helps build muscle. Fiber, as in veggies, also helps keep you fuller, but you need the protein with it.. Fats help keep you satisfied, but make sure you opt for the healthy ones, such as those found in olive oil, avocados, nuts/seeds, etc.
A healthy, filling snack might be an apple with 2 tbsp peanut butter... the peanut butter provides the protein and healthy fats, the apple provides the sweetness and filling fiber.
You might also want to get your thyroid checked to make sure all is well there... hypothyroidism can make weight loss a total nightmare - that's the trip I'm on... additionally, insulin resistance adds more difficulty.. another trip I'm on.. simple blood tests (TSH, Free T3 and Free T4 for thyroid) can eliminate these possibilities.
Peanut butter is good, but make sure it's the kind with no hydrogenated oils, which are trans fat...lean protein and fiber is definitely the way to go, plus lots and lots of veggies. Vegetables are full of fiber