The important thing, I think, is if you ride it when it's soaking wet you will permanently change its shape. If you want to do that, that's fine; if you don't want to change the shape, don't ride it wet. An occasional soaking isn't going to do any harm, but repeated soakings will hasten the aging process.

The chalky look that you mention is, I suspect, something like the white ring a wet glass will leave on a waxed tabletop. The water seems to bond with the wax, temporarily -- I can't explain the chemistry of that, though. That would make sense if I waxed the saddle top... but frankly I don't remember if I did.