And I have never had a problem dining alone in any Disney restaurant. I'm usually pretty oblivious, so I have never noticed anyone staring at me or anything like that. Sometimes, I take a book to read.
On my last trip, I ate solo at Le Cellier and had an absolutely wonderful meal. My waiter was very attentive and took good care of me and I really enjoyed the food.
I also ate at Boma, which is a buffet, but there is a waiter who brings drinks, extra napkins and explains the set up and foods for you. Again, the service I received was just outstanding. I had a nice table next to the window so I could look out on some nice gardens (alas! no animals!).
I ate at 5os Prime Time, which is one of my favorite places to eat at WDW because of the theming and atmosphere, and as usual, I had a great time.
I had dinner at Hollywood and Vine as part of the Fantasmic package, and enjoyed their buffet. The server was very attentive.
If you're worried that you'll be forgotten because you are dining solo, I'd suggest trying a few buffets. That way, you get your own meal and are less dependent on a waiter. Personally, my one exception to the buffet suggestion is Biergarten. As much as I enjoy German food, being seated at a table with a bunch of strangers is not my idea of a getmutlich good time.
Don't worry about what other people might be thinking. Just enjoy yourself.