I live in Vancouver, BC but my trainer is on my covered patio to avoid this very issue. You could pick up a larger dehumidifier, I'm guessing you would need one 30-50 pints if you are having issues to the level you are describing. Do you train in a closed room? Smaller enclosed air space will allow you to get away with a smaller unit. This may not be enough for very heavy sweaters (like me) but could help in your situation.

Dehumidifier?

I don’t really know the answer but I can share my experience. Shoulder season is the toughest part. As it gets a little colder outside, the heated air inside will be relatively drier and things will hopefully be a little better. Maybe somewhat paradoxically you could try to heat the space where you are working out, especially during the time when you are not actively working out. When we lived in Seattle, we had a partially finished basement where we worked out on the trainer and the treadmill and moisture was rarely a problem. The window frames did have some areas of rot but they were also 95 years old and leaky. During the biggest fall is storms occasionally water ran under the door. The basement was all open and had forced air heating from a gas furnace. Temperature wise it probably stayed around 66 in the winter and with a fan that felt good to us for working out. Watching TV generally was accompanied by a blanket. I think we benefited from increased air circulation and a furnace that ran a lot to keep up the with the leaky windows. Now we live on the dry side of the hill and our workout room is an enclosed 8’x16’ space and the heat and humidity feels like it builds up in there so much more quickly.

Only real answer I've found is to wear a long sleeved top and a toque, preventing most of the perspiration from getting into the air in the first place. Oddly enough, my heat adaptation generally peaks right about mid January most years.

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Tech writer/support on this here site. FIST school instructor and certified bike fitter. Formerly at Diamondback Bikes, LeMond Fitness, FSA, TiCycles, etc.
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How about riding outside in your yard or patio? I ride outside year round here so can't offer inside advice but it never really gets too cold so maybe riding outside is possible. You said you have a house.

Use a towel to put as much sweat away as possible. If you use a fan I would place silica gel in some sieves downstream of the turbo. If the silica gel is saturated you should be able to get the water out by putting it in the oven at >100°C.

+1. Buy a dehumidifier - they work really well unless the area is huge. It’ll store the collected moisture in an internal tank or you can connect a water hose to the back and run the water outside, depending on how your room is set up.