It's been a while since I was here (and I see a lot of familiar names!), but this is a subject I can speak on. I've been on one WDW trip solo and one where I was solo for half of the trip. I always do a mix of sit-down and counter service meals while I'm there. If you're unsure about dining alone, go out for a solo meal at home to practice. There are some restaurants that lend themselves to solo dining more than others. For instance, watching the "movies" at the Sci-Fi Dine-In can keep you entertained while you eat. I sat in the front seat of a car with a friendly family in the back two seats and had a nice conversation with my server. I even conned him out of one of those cool frisbees they give to the kids! (However, the food wasn't really worth the price.) I also had very friendly, attentive servers at the Brown Derby and Tony's Town Square. I usually try to avoid the busiest meal times, coming in for a late lunch or breakfast or an early dinner.
One piece of advice, though. If they put you in an inferior table when there are other, better tables available, definitely speak up. This happened to me at a certain restaurant at the Boardwalk which shall remain nameless, I didn't speak up, and I have regretted it ever since. Regardless of how many people are in your party, good service is good service.
My other main tip is to take breaks! Since I didn't have to consult with a travel buddy about what to do next, I didn't stop as often. After a couple of days, I was too pooped to go on by early afternoon. I spent at least two evenings sacked out in my room watching TV and eating counter-service take-out. Not a terrible way to spend the night, but not what I planned either.
Most of all, enjoy yourself! Talk to people. I've encountered some really nice people at WDW. Ask people to take your picture. The main benefit of travelling solo is that you get to be completely and unapologetically selfish. So do whatever it is that makes you happy.