We also do the beach towel thing
! I would also suggest one of those baskets on wheels; I purchased mine years ago from Carol Wright Gifts. While it's not practical for hard luggage, it's a life saver for groceries and the WDW souveniers at the end of the trip. Small duffel bags will fit well too.
Keep a roll of paper towels closest to the non-driving adult, to be ready for any spills. A few plastic grocery bags are a must for garbage. Keep ahead of your trash! Two of them put together makes a decent emergency barf bag.
(yes, that comment comes from personal experience).
Also, allow each child a small bag to keep with them in the car that holds their electronics, toys, activity books, etc. I also have a large tote bag type purse to keep with me at my feet. It's not so big that I don't have leg room, but it holds my wallet on a strap, puzzle book, a paperback, phone, wet wipes, tissues, makeup bag, a small medicine kit (just a few of various pills and other meds), sunglasses, feminine products.

Keep a roll of paper towels closest to the non-driving adult, to be ready for any spills. A few plastic grocery bags are a must for garbage. Keep ahead of your trash! Two of them put together makes a decent emergency barf bag.

Also, allow each child a small bag to keep with them in the car that holds their electronics, toys, activity books, etc. I also have a large tote bag type purse to keep with me at my feet. It's not so big that I don't have leg room, but it holds my wallet on a strap, puzzle book, a paperback, phone, wet wipes, tissues, makeup bag, a small medicine kit (just a few of various pills and other meds), sunglasses, feminine products.