Watch the moon and let your eyes adjust
The moon would be a little annoying watching the meteor tonight, but it shouldn’t damage the screen. If possible, just get a barn or rocks or a mountain between you and the moon so its light doesn’t catch the eye.
Remember that your eyes take 20 minutes to truly adjust to the dark, so give yourself some time. Meteors also often appear in waves, so don’t give up if you don’t see anything after five minutes. They will come!
Also, don’t want to look at your phone once you’re in position, every time you glance at it, your eyes take time to adjust to the shadows and You can miss the dimmer meteors.