Why is my GFCI/Switch combo working in such an odd way?
It sounds like your electrical box for the switch does not have the power source (line), only a 2-wire cable to handle the switching for the light; the line comes in at the fixture. 

Unfortunately you cannot add an outlet to this configuration as-is. You will need to bring a third wire down into the box from the fixture that will act as a dedicated neutral so that you can have the outlet always powered, and then optionally power the light via the switch.

Based on what you've described, my recommendation would be to bring in an electrician to handle this task for you as it requires pulling a new cable from another junction box and correctly wiring it up.