The stickiness of cards is typically just due one of two things, the surface of the card becoming rough or warped due to actual wear, or the build up of oils from contact with the fingers. In the case of oils, which you can tell thats what it is because the cards still have a reflective surface, you can just use a damp rag (not wet) with soap on it, and give it 2 fast light swipes on each side, making sure to hit the corners and dry it off with a paper towel afterward. You can also just take a hair dryer to them, in a spray paint motion where you never stop moving the airflow, then wipe them with a dry paper towel immediately.
I don't know of any products to return them to slippery state, but most games you can buy replacement parts for through the manufacturer, right down to individual cards that may have been damaged or destroyed.