The dampness is more than likely caused by the temperature differences of the concrete to the moist air causing a condensation on the concrete. As far as coloring the concrete , you have several options. I'll list them:
Reactive acid stain: This color process is created by introducing metallic salts to the free lime in the surface paste in an acidic state. The chemical reaction creates the coloring effect. It is the most difficult to control. You must neutralize the surface and wash it real well.
Solvent based dyes: Fine particles of colorant penetrate the surface and color it. Most actual solvent dyes are not UV stable. This process produces vibrant colors often semi translucent.
Water based colorants: This is starting to gain popularity as the chemical makeup of these colorants are improving. They still mostly lay on top of the surface much like how paint does and are easy to scratch off on a smooth surface.
You still need to apply a protective clear coat. I will list them from the least to the best in performance.
Water based acrylic sealer: This product adds protection but is easy to scratch and must be applied in several thin coats. It does not pop the colors as well as other sealers.
Solvent based acrylic sealer: Better than the water based acrylic product. You need to be aware of the chemical make up of the sealer, UV resistance, non-yellowing. More isn't better with either of these acrylic sealers so they must be applied in thin coats. Solvent sealer must be used with great caution on interior applications due to the fumes and potential explosion hazards.
Water borne epoxy: As we start getting into the epoxy based coatings, they offer better scratch and chemical resistance. this product is still a thin mil application.
Solvent based epoxy: This product is the most widely used for seamless floor application. There are different grades of epoxy. The higher the actual epoxy resin content the better the product. Clarity, ease of use, pot life, shore hardness, tensile strength and compressive strength are just a few things to compare epoxies by. There are differences in product brands. The stronger the smell of epoxy, the more it has been cut and diluted.
Urethanes: Mostly used on top of epoxies because they offer better scratch resistance. Some urethanes are stand alone products meaning you can apply them directly on concrete. There are water borne and solvent based urethanes.
There are plenty other types of products but the one's listed above are what the residential customer see's most often. Most of the other products are for the industrial-commercial market.