Skip the free transportation; hire cars or walk instead
Disney transportation, which is offered free to resort guests, is ground zero for encountering a sea of children. And taking the monorail, boat, or tram to reach the gates ensures you'll come face to face with more young ones than at the local daycare. Whenever possible, lace up your Rockports and trek there -- Epcot and Hollywood Studios are actually walkable between each other, and tuckered-out little kids probably won't do -- or utilize hired cars to move around the grounds in privacy.
For Epcot, get dropped off at the Beach Club Resort and walk to the International Gateway to enter; and for Magic Kingdom, try to get dropped at the Contemporary Resort, which has a little-known walking path between the hotel and the park. (Animal Kingdom and Hollywood Studios have designated drop-off/pick-up sections, so use those.) Uber is much cheaper than WDW taxis, but the latter are ready and available -- so take your pick.