Can damp cause serious health problems

YES.... dampness causes mould growths ,bacterial strikes as well as punctures growths , and skin rotting rashes..
Inhaling mould fragments or spores can inflame the airways, causing nasal congestion, wheezing, chest tightness, coughing and throat irritation. Prolonged exposure to high levels of indoor dampness can reduce lung function and cause chronic health problems such as asthma

watch out for black mold it can be deadly

If mold spores are getting into your lungs, yes.

I used to have an EP friend who lived in his mother's basement. They got flooded, and I sent him a dehumidifier. Yes...I was that stupid. Living in damp conditions is not healthy.

Folk wisdom says so, and I have a feeling it’s right there.

Lots of problems

Mold can be a real issue.

Yes. Respiratory issues. Headaches. Nausea. Etc.