We paid all of our food expenses, including tip on the magic bands, and then I settled up at the hotel each night using Disney Gift Cards (Thank you, Target RedCard discount!!!)
As far as budgeting, most of the menu prices are available online through sites like TouringPlans - so when planning our big trip last year I made a spreadsheet (I'm dorky like that) of where we would likely eat each day, looked up the prices and picked what was most likely for my family to order. I tallied up the local tax as well as the tip, then totaled it so I could have a pretty accurate idea of food budget. We ended up UNDER our food budget and had extra money for souvenirs.