A gas turbine that includes: a combustor coupled to a turbine that define a working fluid flowpath; a compressor discharge cavity; a staged injection system that includes the forward injector and a staged injector; a stator blade positioned extending across the working fluid flowpath between an inboard sidewall and an outboard sidewall. A one-way continuous coolant flowpath that includes: an intake section that comprises an upstream port connected to the compressor discharge cavity and a downstream port formed through one of the inboard and outboard sidewalls; an outtake section that comprises a downstream port connected to the staged injector and an upstream port formed through the same one of the inboard and outboard sidewalls; and a cooling circuit extending through an interior of the airfoil of the stator blade and connecting to the downstream port of the intake section and the upstream port of the outtake section.