Your old Outlet might have more than two wires. It might have power coming in and power heading out to the next outlet or light in the circuit. Four screws on the outlet might be being used. You either have to connect all of them to the “Line” side of the GFI(Black to Black and White to White) or else you will use the “Load” side of the outlet to protect everything down stream as well. Connect the black wires to the smaller prong side of the outlet. And it will not work if you do not connect the incoming power to the line side to the GFI.(if you switch the line and load wire sets).
I believe that you can by code actually replace all downstream GFI protected outlets with regular 3 prong outlets. You just have to put that sticker on them.
Pretty cool.
Thanks for the additional information about the connections (and I’ll use the sticker).