My experience is that this is always caused by moisture behind or under the concrete. You may need to install a French drain and sump to correct it, or it may be that you have to expose and waterproof the basement walls, or both.

Black mold on the inside suggests an urgent need to tear down the wall in question and see what's behind it. Do it for health reasons if nothing else. It could be inadequate insulation, failed vapor barrier, or poor caulking has allowed cold air infiltration.

Isn't there another house that would interest you? I love old houses. But if you aren't ready for it, the challenge stops being fun really fast.