Hey there Lisa, I understand your dilemma completely. Being a local, we often just head over to Disney for the day, without making dining reservations in advance. Previously, some "walk-in" tables were always kept for this case at most of the restaurants, but lately and with the addition of the Disney Dining Plan, they seem harder to come by. That is not to say they don't exist, and if there is space, you will be able to dine there.
Popular restaurants like Le Cellier in Epcot's Canada or Cinderella's Royal Table will not have walk-in availability 99.9% of the time. Places like Germany's Beirgarten, China's Nine Dragons Inn, or Hollywood & Vine at the Studios are
usually where we can get a walk-in last minute. It really depends on the crowd levels and time you are trying to find a table. There is also usually availability at
most resort dining locations but heading to a hotel other than your own will add travel time to your day using Disney transportation.
My advice would be to definitely try to find one or two must-eat places that you really want to experience and book
Advanced Dining Reservations for those places. Then just craft the rest of your day's itinerary keeping that in mind. So if you are dining in Epcot at 6, make sure you are either already in that park, or give yourself enough time to get there for your reservation.
Of course there are always counter service options if walking in to a table service location doesn't work out during your visit. Some of my favorites as far as non-cheeseburger and chicken nugget fare go are the Sunshine Season Food Fair at the Land pavilion, the ABC Commissary at Hollywood Studios and the Columbia Harbor House at Magic Kingdom.
I hope this helps with your planning a bit. If you have any questions about specific restaurants or anything else, just let us know.
~Suzannah