L.R.
Definitely sounds like allergies -- Good that you checked for mold, but you may need to actually take out some of the drywall somewhere to see if mold's behind it. Age doesn't affect whether the house has mold, any age house can get it. Check online for companies that mitigate mold and see if you can get advice from them on whether it sounds like mold or not.
This also could just be sensitivity to very dry winter air. Running our heat dries out our houses badly and the nasal passages can really suffer. Try cool mist (NOT warm mist!) humidifiers in your bedroom and baby's all night and start them a few hours before sleep time. This helps a lot with nasal issues.
Also, have you ever had the heating/AC ducts professionally cleaned out? Companies come and suck out everything in the ducts. Can't hurt. Be sure that whoever does it covers every vent inside the house or you'll get dust pushed out through the vents -- making things worse.
And see a doctor for you, not just for the baby! Good luck and feel better!