jb suggested insulating the floor. This would be worthwhile even if not required to solve your problem - to save heating costs.
The cold floor is a suspect. The ceiling heat warms the top of the rug, but not necessarily the floor underneath. A floor can become cold enough to drop below the dewpoint, so the wood and rug pad could absorb moisture. The ceiling heat would release some of the moisture. Since there is no ductwork to pierce the surfaces to lose air between joints, the moisture level would be higher in the tighter house.
Insulating the floor would raise the floor temperature enough to prevent the moisture settling on the floor surface. Use unfaced batts so moisture can pass out of the fiberglass instead of settling on a cold paper or plastic surface facing toward the ground. Insulation spring rods are available for underfloor installaion.
When you do this, insulate the exposed pipe, as the loss of heat to the crawl may bring the temperature below freezing. (Which would also trap the moisture as frost on the ground, removing it from the air in the crawl.)