Thank you, everyone, for all the helpful suggestions! I think my solution is going to involve a bit of everything.
I ordered a small dehumidifier, and some damp rid type stuff. I picked up a tiny heater at a thrift store. I'm running my big heater, with a window and the roof vent open, for about 6 hours a day, some in the morning and some in the evening. I'm keeping the cabinet doors open. The weather has improved a little, but not a lot.
The insides of my cabinets are finally dry!
The weather has improved a little, but not a lot. I think the thing that tipped me over into problem land was that I tried to dry my wet rain clothes, and a couple pieces of wet laundry, in front of the heater with the roof vent closed. Now I know better.
Long-term, I might make some structural changes to the cabinets. And I was planning to build more cabinets next summer, so now I know that I need to incorporate air circulation into the new ones!
