Those conditions are exactly why IMHO anyone who wants/needs a fire brings a small saw big enough to cut a piece of downed wood that even if wet will have a dry center (heart).

Something to split it and initially a fire starter of some kind to get things going.
Birch bark is great and If you run across some, it's a good idea to stash some in a zip lock.

I'm partial to my paraffin wax/ dryer lint/ cardboard egg carton homemade firestarters.
I've added a trick b-day candle center wick to give it a little extra bang.

The magnesium powder in the candlewick makes these burn hot/fast, enough to light even the wettest wood, but the dryer the fuel source the less smoke, although if the bugs are bad, excess smoke might be a good thing..