I think you will mostly need to make substitutions (ex sub veggies or sweet potato for mashed potatoes) or eliminations (ex order entree salad without cheese). Though, I am sure many of the salad dressings contain sugar, so not sure if you are willing to flex there. The Colony Salad at Liberty Tree Tavern and the Chicken Strawberry Salad at The Plaza Restaurant are two of my favs, but again, not sure of the contents of the dressings. You may do well at some buffets. I am also thinking Ohana may be a good choice. You could skip the bread, noodles, and bread pudding if you wish (however if I was going to make diet exceptions I would make them here... Yumyum) and still have salad, cooked veggies, steak, pork, shrimp, and chicken (all in endless supply). Garden Grill, which is family style character dining, offers a similar situation with salad, garden veggies, fish, turkey, and beef. You would have to skip the bread, dessert, and I think you may be able to sub sweet potatoes for mashed here. Lastly, check out the menu for The Wave (Contemporary Resort). They have several meals that I believe fit the bill exactly, and possibly even a dessert or two that may work for you.
I will try to think of some quick service options and report back later. My family is not Paleo, but we follow a diet that is along those lines. I find it pretty hard in Disney, and we end up making a lot of exceptions. I try not to sweat it too much because I am finding that spending a yearly vacation in WDW with my family contributes greatly to my happiness and well-being. But I do wish there were more yummy health-conscious, organic whole food options available.