I have been playing once a week for a month. Is that enough to improve or should I play more? Worried about overdoing it too.

Once a week is a great start! At the beginner stage you improve fastest by playing 2-3 times per week. Your body needs time to build the specific muscles used in badminton. Listen to your body though - if your shoulder or knees hurt, take a break.

I went from once to three times a week and the improvement was massive. But make sure at least one session is focused practice (drills, footwork) not just games. Playing games only builds bad habits.