C.Ann, don't feel awkward about going solo. Going solo most definitely does not mean you are "alone"! You're in a vacation kingdom sfilled with tens of thousands of happy people.
When I take a solo trip, I don't feel the least bit awkward about eating alone in the WDW sit-down restaurants, especially those in Epcot's World Showcase, because the CMs will often make a special effort to talk to you and make you feel at home.
That being said, I have found certain places where I am most comfortable dining alone. 'Ohana is just perfect, especially if you get a table in the back of he dining room. I think 'Ohana would also be very romantic for couples looking or a relaxing getaway. Boma is also terrific, if you get a table to one side of the dining room away from the buffets. I found Concourse Steakhouse very quiet and relaxing when I ate there solo in December. Boatwrights at POR is great for solo diners (I have only eaten breafast there however). And if you make it over to Universal, Mythos in IOA is a fantastic place to eat.