What brand and model is the new switch? Different models need to be wired in different ways.
The wiring FAQ may help:
(BTW, for technical reasons, there are no GFCI zwave devices, the two requirements are incompatible, so that’s not the issue you’re seeing. Beyond that, although GFCI light switches do exist that aren’t combos with a receptacle, they’re pretty rare, typically only used for bathroom fans. In the US, there’s no national code requirement for GFCI wall switches like there is for receptacles (although the manufacturers of damp-rated devices may require them, which is where the bathroom fans come in). Switches require AFCI instead, but that’s the same for both networked and non networked devices.)
Given that you’re unsure what the load screw is for, I’d suggest bringing in an electrician. Or if you live near a Home Depot, many offer classes in how to install a light switch which are pretty good and should at least keep you from burning down the house or shorting out your brand new switch. 