U.S. State Government Adopting Parliamentary System

[I Am Feeblepizza.]

Yes, it would be constitutional due to it being a republican system. Besides, there are already plenty of weak governors out there. In Texas, for example, the lieutenant governorship is a better position than the governorship when it comes to pushing programs through the legislature. In Indiana, the legislature can reassign statutory power from the governor to other department heads, rendering the governor essentially powerless. So it's not that much a stretch, really.