I dine solo all the time at home and I have done it in WDW many times. It rarely bothers me to be in a restaurant alone. However, one time I ended up at Hollywood and Vine for lunch and the Playhouse Disney character meal was still going on. I thought my ADR was late enough that it would be over but sure enough I found myself surrounded by rowdy preschoolers and characters. The characters kept coming over to me and clearly didn't know what to do. They pointed to the chair, shrugged their shoulders, waved awkwardly, etc. It was the only time I have been uncomfortable dining solo in WDW. I ate as quickly as I could and got out of there.
Since then I have enacted a personal policy of not doing any character meals solo. However, I have truly enjoyed my solo experiences at Le Cellier, The Wave, and Hollywood and Vine again but only at dinner time. On my upcoming trip I have solo trips planned for The Wave breakfast buffet, possibly Hollywood and Vine for dinner, and maybe one or two more places. If you don't mind weird interaction with characters go for the solo dining at those places. If you think the potential for awkwardness would bother you, consider some other ADRs.