Shrimp and crabs can stay in the tank but every fish must be moved out for this process to work.
a tea towel or normal bath towel is fine to cover the bucket with. Make sure that no part of the towel can slide into the bucket otherwise it will allow the water to drain out via capillary action.
Also make sure the buckets are not sitting on the floor, especially tiles or concrete because it will cool them down. Put them on a stand, table, shelf, or even just put a couple of books under them.