I use a dry paper towel because that's all it needs. It doesn't NEED a wet one, but a wet one won't hurt it either.
If dampness was so bad for meer, they wouldn't make smoking pipes out of them—the moisture from the tobacco would kill them faster than a damp paper towel.
I don't wipe mine immediately after a smoke, but I wipe it immediately before the next smoke. I've never felt the need to do any deeper cleaning of the chamber than a dry paper towel. No cake build up. However, I also don't smoke tobaccos that might ghost them.
And for what it's worth,
@Chasing Embers recently deep-cleaned one with an alcohol soak, proving pretty unequivocally that you're not going to melt or ruin a meer with cleaning.