I would NOT flush paper towels. That is asking for trouble.
If it were me in a apartment I would just use a plastic bag from a store and throw it in the dumpster. Kinda like what you would do with a dog.
However, In my snake room I have its own trash can and it has never made the room smell foul. When the snakes have freshly soiled their cage it does but not once its in the trash can.