To dry an object, as you know you want the water to evaporate off of it and out of it. Heat is one of the obvious ways to do that.
The dumbed down chemistry lesson
However, if you increase the humidity of the air, you will indeed slow evaporation and therefore slow the drying process. The water molecules leave the object you want dry when some of the surface ones have enough energy to break free from the rest.
To drastically over-simplify it, imagine the molecules as a bunch of sticky balls all vibrating around (think maybe one of those kid ball pits, but all of them extreme bouncy-balls, covered in slow-drying glue, and with a motor at the bottom keeping them all vibrating), vibrating hard enough that they are moving relative to each other (ie: not stuck in place like a solid). The ones that vibrate hard enough can break free from the sticky mess entirely and get away. The water molecules that do that become part of the water vapor in the air. When water boils, that is when all of them are on the verge of breaking free, and many cross that boundary rapidly. For simple evaporation, you have to wait as the energy randomly shifts around in the water until one of the molecules happens to get bumped around just right to have that kind of energy.
Back to the stick-ball analogy: imagine that a bunch of other balls have already broken free of the mass and are wildly zooming around through the air because of all the energy they have. The balls trying to leave the sticky mass will have a hard time doing so if there are too many balls ready to knock it back in. That's kind of like how evaporation works when the air is too humid. The balls (or water molecules) can still get free of the mass into the air, but they now require even more energy to ensure that it happens at a reasonable rate.
The ramifications (and crux of the answer)
So if you raise the air temperature just a little bit but raise the humidity a lot, you're much worse off. In fact, I would suggest that keeping humidity as low as possible is more important than adding heat, assuming the heat is in small quantities of only a few degrees, or even a few tens of degrees. If you can raise the temperature while also keeping the humidity constant or lower, then that will help, yes.
If you can increase the heat a lot though, then it is worth it. If you can increase the temperature of the room to a significant fraction of boiling temperature (close to 100C or 200F), then you're good... except that then it's way too hot, so you're not good. @JiK suggests in comment that some people do tolerate this temperature for extended periods of time, but you won't catch me sitting in a 200 degree F room for an extended time. I certainly could not handle that.
So the answer, as long as you are willing to tend the drying process very carefully and actively, is to dry with a flame and get your fabric close enough to the flame to heat up a lot. But only the wet portions; you will ruin the fabric if you get drier portions near the fire too, or if some of it dries out before other parts and is still too hot.
What I do sometimes is to hold fabric close to a fire, just out of reach of burning myself, until the water starts to come off fast and visibly as steam, maybe even making a sizzling sound, then I pull it back. That dries it out a lot faster. As for using propane, this might work with a propane stove, but I would be wary of that and way extra cautious.
Keep in mind that as you are drying your fabric, the water coming off of it will itself raise the humidity around it. So if you are in a small, enclosed area, each unit of drying will be more difficult than the unit of drying that preceded it.