I never liked the wet sponge.
Using cool water randomly on a metal surface with a delicate coating does not sound like a very good idea (perhaps only for the guy selling the tips
)
I use the cheapest tissue I can get to clean the tip. I also have some of that dry brass sponge, but it seems to lose its effectiveness fast for its price. I still have a couple in their plastic bags... I do not care to use them. Tissue works fine, is dirt cheap and I use it already around the house so no need for extra supplies.
And I put a nice blob of solder on the tip after use. The flux burns and loses its activation agents quickly so no reason to worry about it.
If it did not, then it would oxidize the pcb too.