look in your connectors and make sure nothing is jammed in there like dirt or something to prevent a connection. A bad connection from the connector on the device will not be fixed with a new cable.
as for replacing the port, it would be very difficult to just change it out. maybe you can see what hole has an issue by looking for a lack of conductor inside them.
DVI runs in a few different modes with different pinouts, it sounds like your on a DVI-A for an analog signal but I havent see what the DVI-D signals do with limited pins connected.