If I understand the new proposal you will still have plastic bags available you can recycle, you will still get newspapers delivered in plastic, some grocery items will still be put in plastic, it is _only_ the "checkout counter" bags that will be most affected. This is a great proposal and good for Floridas ecosystem.
It is good you recycle the bags and you are to be congratulated, most people do not recycle the bags. The average "used life" of the overwhelming majority of "checkout bags " is estimated to be about 12 minutes ( 6 minutes from the checkout counter to the car, 6 minutes from your car to your kitchen). Since you are creative and already a recycler your transition to using less "checkout counter" bags will raise your awareness to the multitude of other bags already available, just a quick inventory of my kitchens bags: the waxy bags inside cerial boxes, bread bags, rice bags, popcorn bags,pizza shells, pretzel bags, potatochip bags, produce bags (grapes etc) newspaper bags, lots of items prepackaged in bags, many items bagged inside of boxes. Bring your cloth bag shopping, no big deal !