Nope, no danger of wrecking the engine. Default position in the event of runner flap motor failure is flaps fully open. If the flaps were stuck closed you would know about it as you'd have no power past 2000rpm.
Try a basic settings/adaptation in VCDS for the flappy flappers a few times and check you get "OK". If no OK, the motor is buggered. When doing that test, you should see the motor actuator arm moving.
10 days to go flat isn't bad tbh. If it went flat in 3 days then you definitely have a duff battery or an alternator charging issue. If the battery measures more than 12V with the car gone to sleep, I wouldn't waste money on a battery.