What are my options to fix cold and drafty floors over a vented crawl space?
Cap the floor joists with rigid insulation. I'd first make sure the existing insulation is completely filling each joist bay and as expanded as possible (no edge curls, gaps nor compressed sections). Remember, the paper vapor barrier should be facing the living space you are heating or cooling. For a truly draft-free seal there is now a spray foam kit sold for the DIY home owner. Same 2 part spray foam that is used by commercial insulation installers, but in smaller amounts.