We are going end of May, beginning of June so we may have left it too late for some places.
And I'd say i'm picky - a vegetarian that doesn't like most vegetarian food that's offered!!
Also, as we are travelling from the UK, I wouldn't know how to go about changing or making reservations over there - we tend to leave our mobile phones at the hotel as they are added bulk and too expensive to use abroad.
You can make ressies online and change them online as well. Go to
www.disneyworld.com to get started. Or, of course, you can call. Changing reservations (or making them if you haven't yet) while at WDW involves simply going to any guest services location (park, downtown disney, or any WDW hotel) or picking up a resort phone. Or, of course, you can still do it online or using your mobile.
Late May, early June isn't an extremely busy time, so you should have some choice left on ADRs, but, as others have said, make them ASAP. Better to have something and decide not to use it then have no ADR at all.
That being said, I have, however, been successful at getting reservations in parks day-of, even during busy times, by being flexible. Have table-service at lunch instead of dinner, or be willing to eat dinner at 4:30 or 9 pm. Or be willing to take whatever restaurant in the park has availability. Over the December holidays, some combinationof Nine Dragons, La Hacienda de San Angel, Biergarten, and Restaurant Marrakesh, generally had same day ADRs available at Epcot. Tusker House did at DAK. And Mama Melrose's did at DHS. MK was much harder, although we did snag a same-day ADR at Crystal Palace, which was extremely lucky.
If you expand your table-service locations to the entire resort (hotels and DtD), it is much easier to find ADRs, although, of course, the better and more popular places do get booked early. Some places that we were able to get ADRs with little advance notice, even during the Holidays, included Kona Cafe, Trail's End Buffeteria, Yachtsman Steakhouse, Cape May Cafe (although that was a shock), Captain Jack's, Sanaa, Wolfgang Puck, the Wave, Maya Grill, Rainforest Cafe, and Fulton's.