Stuart, you haven't said what make and model your bag is, does the shell (that's what the makers calls the outer material) have any sort of water resistant treatment? Furthermore, some bags have treated down as well (hydrophobic is the word) that should be less affected by damp.
Mine has some sort of a finish on it, it's a Vango Venom 600, 2015 model. I'm quite neurotic about getting any wet on it, though this has happened once or twice, just the odd spot, and to my great relief it hasn't hurt at all.
In essence it means that the sort of dampness you describe isn't going to harm the bag, it won't (shouldn't) get through to the inside. I've experienced it myself, it's like a sort of dew that settles on everything inside the tent including your sleeping bag, usually when the air is still and windless, and worse if you're pitched in a particularly damp spot, with everything outside wet from recent rain.
I think your bag would only be harmed by getting it wet inside, and putting it in the tumble drier shouldn't be necessary. It should be sufficient just to hang it somewhere in a room indoors, or drape it over a washing line if it's either windy or sunny. Remember that even if the outer has remained bone dry it still needs the airing to rid it of the moisture inside that has evaporated from your sleeping body during the night!
The latter point is also important during your trip, you should make a point of airing your bag when possible (taking weather or lack of somewhere to air it, as I mentioned in my previous post, into consideration). It might be in the mind, but I am sure my bag feels more cold and clammy when I get into it on nights when I was unable to air it in the morning.