My skin around my nails is a bit messy while I'm doing my nail art. My trick is to wait for my entire nails to get totally dry then I'll apply some soap and scratch the skin around my nails that has excess nail polish using my thumb nail. It's much easier to remove them when it's totally dry because it's already hard.
I don't use cotton buds with nail polish remover and rub it on the skin around my nails. You may accidentally remove your nail polish on your nails if you're careless like me.
I would usually do my nails before going to sleep so that when I wake up, my nails are already dry, but if you don't sleep like an angel, paint your nails when you don't have any plans for the entire day.