So I always write in the reservation that I want to have a moisture-proof mattress protector.
At larger houses, this is often standard anyway, and if not, it's no problem that they provide one.
At smaller hotels or guesthouses it can be a problem - but then I either book something else or just take a few bed pads with me.
From my experience, the washable ones are only suitable to a limited extent. Especially if you are only one or two days there, you often do not get them dry if something has gone wrong. This is then possibly avenged at the airport if you are only traveling with hand luggage and the scanner finds the wet thing ...
You do not have to be ashamed at all to ask for something like that. You are not the only one with such a problem and the hotels are usually prepared for it.
In this context, opaque garbage bags are also useful, because the trash cans are usually too small.