Yes, it's VERY possible that the warm air is drawing the moisture up from the damp crawlspace. This would ESPECIALLY occur when the floor gets heated and eventually cools. (Remember, cooling need only be 'relative' to the air temperature. If the room had a daytime temp of 72 and it dropped to 68, that in effect is cooling and will PROBABLY cause moisture to condense out of the air. Assuming a dew point of 68 or higher ...)
Start to solve your problem by insulating the floor as I mentioned in my previous Follow-ups. You could try to use a dehumidifier but that will only draw up more moisture, and give you a new 'job' of emptying the bucket! It's VERY possible that the heat in the house is 'heating' the crawl. (I know I'm repeating myself, but ...) If the outside air in the crawl is 30 and it heats up to 42, then cools to 30, you'll get condensation and then frost. Yes, this will draw moisture up from the ground and it will condense out on the UNDERSIDE of the plastic. Use R-30 Kraft-sided insulation with the kraft side facing the HEATED room. Review my link from my past Follow-up.
If you need more 'guidance' e-mail me directly. My best to ya and hope this helps.