I don't know how it works at your location, but around here, cable is separate from phone ASDL...and never the twain shall meet. Typically, cable is provided by only one service via contract with the community being served. If you still had ComCast when the CenturyLink installer came out, he may have set it up to access CenturyLink via ComCast's cable service which would mean the CenturyLink connection would depend upon the ComCast cable service. A dumb way to do it, if that's what happened and CenturyLink owes you a free service call to correct it and, as mentioned, an ASDL modem (rent or you buy).
One way to determine if this is what has happened is to look at your current modem. Besides the fact that it should have some identifying information on the label, if it is a cable modem, chances are the incoming line is a coax. If it is an ASDL modem, the input line will be a phone line. If it is coax then it is still depending on ComCast service.