Damp is to mold what weeds are to snakes. A field completely overgrown in weeds may not have a single snake in it. The field next door might only have a small patch of weeds where three snakes call home. The only way you will ever know is by testing.

I take it to be a finished basement.

To get rid of the smell, you have to get rid of the source of the smell.

That would be the carpet, any padding if there is any, and remove the drywall, and probably the firring strips the drywall is attached to along the foundation walls.

Years, if not decades, of stank have embedded themselves in everything.

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When I moved into my house, the basement had a rank odor to it. Shortly after moving in I had three dysfunctional rooms demoed. Smell in basement much better. Recently had another room demoed out. Basement smelling pretty fresh.

All that's left to demo out is a sheetrock ceiling in the main portion.

I also run a de-humidifier. It pulls close to a gallon of water out a day on average.

My guess is that the carpet of below grade concrete is the culprit. A house of this age would not have had a vapor barrier below it. Pull back the carpet and look for signs of mold. Tape down a square foot of heavy plastic for 24 hours and then pull it up to see if condensation is on the bottom. If so, I'd get rid of the carpet and go for some form of hard floor. There are, however, some measures can be taken if carpet is desired. the floor can first be sealed with "RedGard". This is a heavy urethane based primer which will block water vapor. Also, there are special fibers used n many carpets for below grade that make them more vapor and mold resistive.

There is also a topical mildecide named Concrobium which will aid in mildew suppression. It is totally clear, safe and odorless. It can be sprayed on all surfaces and leaves an invisible mold retardant.