When you are travelling and you stay less than 2 days in the same place, specially in winter time when there are not a lot of sun hours or even rains outside, it's really difficult to dry clothes. Sometimes you can find laundromats that may help, but you cannot depend on it always. So, the question is: how do you dry your clothes when you are continuously moving?
Here are some of the 'hacks' that I've used:
It's winter, there's often a heater on, or heatpump, or fire. Try and hang the clothes near to (but not on) the heater. Turn them frequently - you don't want them getting too hot or burnt.
Use air - moving air. Hang in a doorway for the internal flow, or if possible, outside during the day. Beware to take them in before sunset though - it gets damp quickly again as it cools outdoors.
Use sunlight. Move the clothes to a window or similar where light is shining on them for warmth.
Move the clothes on a rack or item that keeps them aired (ie the less surface of the clothes touching something else, the better).
If desperate, a fan helps move air around the clothes. It also makes things feel colder though.
Are you cooking something? Keep your clothes near the warm oven for that electrical heat.
Do you have a laptop/tablet? Don't do this for soaking wet clothes as you may risk damaging the gear, but I've done this several times in the past - sit nearly-dry socks under my hot laptop, and by the time I've checked email and Travel.Stackexchange I've got warm and toasty socks!
Wear the clothes? This one is debateable - I don't risk it. Yes you can use your body heat to help dry out the clothing, but I've been told it can cause the clothes to get a bit of a funky smell. I've got no sense of smell so daren't risk it - but apparently you can kill the smell with a bit of a vinegar mix in your next wash.
To speed up drying of clothes (particularly if staying in hotels where you have laundered towels), you can wring your clothes as dry as possible, place them (individually) on top of a towel and then roll the towel/clothing article up as tight as possible.
Then stand on this roll. The aim being to absorb as much water as possible out from the clothes into the towel.
This won't get the clothes dry, but it will get a lot dryer than wringing them out will do alone. This will make air/heater drying work a lot faster.
The downside is each towel is probably only good for a few shirts/socks before it's too wet.
(I used this to avoid paying $$$ for hotel laundry fees when travelling through europe last year - they were ridiculously expensive!)
In addition to Marks answer https://travel.stackexchange.com/a/24255/4584 if you have an iron handy, you could iron the clothes and that would dry them too. Although in my experience I have rarely come across a city where I can't find a coin operated dryer.
Well, if you have not something which causes heat like a fire / a radiator / a laptop or at least a warm room, you effectively cannot dry your clothes as you already found out. This is an old problem for travelers in polar or high-altitude mountain regions. Even in low temperatures you are still perspiring and this humidity cannot be removed because it remains in your clothes or freezes at the surface. It could be so bad that you are forced to sleep in your sleeping bag fully clothed because if clothes remain outside they are likely to freeze into a form solid as wood which you cannot put on. So the only way is in fact using clothes which are either functional (wicking action to move the humidity outside) or are still warming even if wet (Meaning synthetics or wool, but be careful because some synthetics get brittle in low temperatures). Cotton is the worst fabric in winter and should be avoided.