Hi, Danelle!
Hoping to get a table without an Advance Dining Reservation is a real gamble. Sometimes you will be fine as long as you are very flexible with your locations and your times. Other times, you might just be out of luck!
Here is what I recommend: Book at least 1 or 2 Advance Dining Reservations for dinners. Especially if you want character dining or a particular restaurant. Then, how about booking a table service breakfast? This way, you have used a table service credit for the day without interfering with your park hopping, you will have had a full breakfast, you can use a snack credit mid-day then wherever you end up, you can go to a counter service restaurant for dinner whenever you feel like it.
If you are flexible/adventurous eaters, leave your "free" day of no reservations for the day you are in Epcot. With so many countries that have sit down meals, you will likely find one with an availability.
Booking "day-of" will be much easier if you have a "smart phone" where you can go to
www.disneyworld.com and click on Dining Reservations. You can simply choose which park, the date, and general time (ex: dinner time.) It will show you all restaurants available and you can book right on your phone, saving you time from checking in the parks for reservations. If you do not have a phone, you can check at your resort concierge or at any theme park.
So if you are feeling lucky and are flexible, take your chances! You might just find a new favorite restaurant! I would just feel better if you had at least 1 or 2 meals booked, as you want to get the best value you can out of the Disney Dining Plan. If you need dining suggestions, just let us know. Good luck!