Don't sweat it (literally) -- with a little planning you can stay cool and comfortable!
My family walks an average of 8-10 miles in an average Disney day (and that's with touring in two 4-5 hour chunks with a big leisurely break in between and everybody in bed before 10pm). To save your feet and sanity I recommend a lengthy nap/swim break from after lunchtime to about 4pm. All you'll miss in the parks are the worst crowds and heat of the day. When you return to the parks all rested and refreshed (and with a new coat of sunscreen), wear different shoes. Alternating footwear will help prevent blisters and keep your feet in good shape.
Also, choose appropriate footwear that you've broken in at home. Good-quality running sneakers with moisture-wicking socks, or supportive hiking sandals that don't get slippery when wet (e.g., Teva/Keen/Clarks), are ideal. We bring 1 pair of sneakers and 1 pair of sandals per person for my family of four, switching them midday, and we've never had a single blister! If your girlfriend wants to (or you want to) wear some cute fashion sandals out to dinner or something, that's fine, but just keep a back-up pair of "sensible shoes" in the park bag. You'll be glad you did.
As for clothes, natural, breathable, quick-drying fabrics in light colors are the way to go. And while you don't have to start working out or anything, if you make a habit of taking a nice, long walk in the morning or after dinner, you can help build your stamina for park touring.
You also mentioned being uncomfortable from eating junk food. While Disney offers plenty of that, if you want to avoid salty, fatty snacks, you shouldn't have to look too far. Most Disney eateries (aside from the bakeries and ice cream stands, of course!) offer some more healthful options if that's what you're looking for (either advertised, or on request -- e.g., "could I get apple slices with my grilled chicken sandwich, instead of the fries?").