Hi, Kendra! If you don't have a child in a stroller, I'm not sure you're going to need that much stuff in the parks. Besides, since they don't allow coolers as large as the cart you've suggested, I have a feeling that security won't allow you through with the cart. It seems that it might be a safety hazard in crowds. Plus, you will definitely grow tired of keeping up with your belongings throughout the day.
Instead, I suggest carrying a backpack with a few essentials:
- Camera and/or video camera, accessories, extra batteries and memory cards
- Sunscreen, sunglasses and a hat
- Rain poncho(s)
- Bottle of water & maybe a snack
- Cell phone, tickets/resort key/ID/credit card(s)/cash
If you do have a child in a stroller, you can pack a bag that can stay with the stroller when it is parked with things such as diapers, changes of clothes, snacks, and other belongings that aren't valuables. Then, you can have a park pack that stays with you at all times. In addition, you can divide up your belongings among adults in your traveling party, and/or you can rent a locker at the entrance to any of parks to store any of your items. I hope this helps you plan. Come back and ask more questions if this didn't fully address your needs.
Have a wonderful day and a magical vacation!