I have a bunch of styrofoam-lined bags, but don't generally bring then into stores for a few items (I'll change that), but for the long drives to the stores I really like (Green Acres is NOT the place to shop) I use a big cooler loaded with ice.
The downside to that is that whether my groceries exit the store in paper bags or the more permanent ones, rearranging happens anyway, in order to pack the critical items in ice.
Handy hint: stapling the handles on paper bags WITH HANDLES is quick and easy and so much stronger than glue.