You certainly won't starve, but
free dining periods tend to make it difficult to get into a lot of restaurants. Often, many restaurant will not take any walk-ins during the free dining periods. You may be able to get into a restaurant at a very off time (very early or very late dinner, for example), but you will likely be shut out of the most popular places (such as Le Cellier, 'Ohana, Via Napoli, and Be Our Guest).
Since you're there during F&W, this may be less of an issue since you can fill up quite well by trying food from the booths. Another option if you have a Smart Phone is to try for a dinner reservation once you know what park you'll be in that day. You may not get a first choice spot, but there are usually a few reservations left same-day at a few places. The same places with a same-day reservation may not allow a walk-in a few hours later.
Another idea is to make a few reservations at spots that don't require a credit card guarantee. This way, you know you have reservations at a few restaurants you really want to try, but you can cancel them that day if it won't be able to work for you.
Good luck! There's tons of good food at Disney, so you should enjoy yourself no matter what you choose.