Reducing your cable subscription to basic is not cutting the cord. Discontinuing cable-tv altogether (and putting up an antenna) is.
Slingbox HD-pro has a built-in ATSC tuner. For other models, you can add a stand-alone ATSC tuner to the mix.
My gut is that you can't couple the two streaming devices. You may need to connect one of them to an HDMI-to-component adapter.