Another cause could be a worn magnet or drum in the magnet contact area. The magnets tend to get "grooved" with age and the grooves can cause sticking.
The entire brake assembly should be available at less than $40. It is about the easiest repair for time and money involved. You may also wish to check the drum runout to make sure you have a round drum.
I would change the tire immediately. It's paired wheel is not benefiting from having a different sized partner. If your tires are 4+ yrs. old, I would suggest to swap out all four.
It's a bit of expense to get all the running gear right but the good news is that it only needs to be done once.