I 100% agree that your ventless heater is most, if not all of, the problem. The products of combustion of gas are CO2 and H2O. You're pumping all of that water into the air and it's condensing on any surface it can find that is colder than the dew point.
NOW...I have had a problem with an office area (old construction) where they firred out block walls and just drywalled over the firring strips with no insulation. The drywall had a lot of problems because of excessive moisture. The ultimate fix was add 2" of polystyrene on the wall's exterior and then cover that in metal siding. That worked because it allowed the interior surface of the wall to stay above the dew point temperature, so water didn't condense on it.
The idea to use a dehumidifier seems wrong to me. Use an appliance that makes the humidity sky high and then use another appliance to try to fix the humidity? If you get the right applinace to heat with, you won't have the problem to begin with.
I like ventless heaters as occaisional or backup heat. I wouldn't recommend them ever for the sole source of heat.
Phil