I agree with your reasons, and the heat exchanger can cause water issues in a home when it has a leak in it. I'm wondering if your co detector is working.
I am also trying figure out where the moisture is comming from, sometimes it is all things added up to create enough to cause the problem.
From here it is hard to tell you where to start, but the hrv is one solution to the exchange of air for the furnace, not the solution to the issue of the moisture.
The reason I say to get an expert in there is, you may be missing something you just do not recognize as an issue. Some folks seal up the house with insulation, new windows and weatherstripping...but do not seal the attic pull down staircase.
Try another set of eyes on the house, or call some folks with questions. Your local heat folks will come out to take a look at your issue if they think they can make a change, ask there for some advice.
To me this is a whole house solution that needs to be addressed, it may take more than just the hrv.