Weight for 13 Year Old

Im 13 and 5 feet 2 inches. I weigh between 105 and 115 pounds. I’ve definitely got some squishies and I’m REALLY uncomfortable about my body. I am on LoseIt (goal weight is 95 before my 14th birthday in June) and eat really well during the day, but at night it all crumbles down. I try to exercise, but I never have time and it’s always super embarrassing in front of my family. Questions:

Should I change my goal weight?

How do you prevent nighttime munchies?

How do you exercise with no time and no gym card for six more months?