We have one of those round Honeywell models which we bought when they originally came out about 25 years ago. That was excellent at removing dust. When it was turned on you could see the sun beam dust motes in the air being actively pulled towards it. Plus the coal filter did freshen the air. However, our electric bill jumped so we only use that one once in a while.
The ones we use all the time now are the Winix machines which don't seem to raise our bill. We have three of them--one down the basement and two upstairs--running 24/7. Two of them are this
model and one is a smaller, more portable model they don't sell anymore. (I use that one when my mom goes into the hospital. Takes down some of those smells there.)
The filters get very dirty so I know they are working. Plus, if I burn something or someone comes in the house with perfume on, on Turbo setting, it pulls the odors out pretty quickly which helps a lot. (Turbo is loud though.)
Also they have sensors that work fairly well. When I smell something usually I look over and one or both of the sensor lights is on so it "smells" it too.
It has a Plasma Wave setting which is suppose to put out hydroxyls which I think is similar to ozone but we turn that off right away because I've read some bad things about both hydroxlys and ozone. The nice thing is you can turn that setting off.
Also, the antibacterial filters that come with it are more expensive to replace and have that antibacterial chemical smell that gets to us so we just use the two-part aftermarket filters--both the hepa and charcoal. They seem to work just as well without that added chemical smell or the higher price.
These machines would probably cut the mold issue down for you some too but I agree with the others that you will want to address those issues more fully. I use a dehumidifer in our basement. Both that and the Winix help when it smells musty down there.
Also, after you do that you might consider having a professional company come out and clean your air ducts. Make sure they are the kind of company that blocks off all your vents and vacuums the whole system out. Our whole house smelled wonderful after we had that done, plus we were able to lower our furnace thermostat 2-3 degrees from then on. The owner didn't think it would work that way for us but it did. Our ducts must have been pretty congested with dust.
One note: He offered to spray a special freshening sanitizer into the ducts too but we have chemical sensitivities so declined. If you are sensitive make sure you mention this before they spray anything.
Edit: At some point we bought the very expensive IQAir but it has so many layers to it's filtering system that I don't think it can really pull the air through. It pulls the air from the bottom and then blows the clean air out the top but when I put a piece of typing paper underneath it to test it, it couldn't even lift the paper so we sent it back. It didn't really do anything and was soooo expensive.
Edit #2: This seems to be one of their latest Winix
models and seems less expensive. Plus Fellowes had an identical model so I think their purifiers work on the same principle with a very similar filtering system. This
model by Fellowes automatically adjusts the fan speed when it detects something. I'm not sure if that would be a good thing or not.
Anyway, sorry for the long post. Hope some of the info is helpful and not too overwhelming.