When I was in school, I used to go to WDW regularly with a friend and his dad. After school, I didn't go for years, then just decided to suck it up and go alone one year. I was worried I would hate it - but that first solo trip was amazing. And I've been back frequently solo since - just as amazing.
I'd agree that Tea Party is probably the one ride that would be odd solo. I'm not a fan, so that's not a problem for me.
As a solo over-30 guy, a few of the more "kid" attraction seem odd, as well. I went into Enchanted Tales having no idea what it was, and then felt a bit awkward when it was mostly parents and kids. Having said that, I helped a couple of folks take pictures, and it didn't seem as weird. (I felt almost like a CM at that point, which was kinda cool.) I don't know that I've ever done the stuff in and around the former Mickey's ToonTown / Storybook Circus.
But the CMs are used to seeing solos. Sometimes, I think they're more chatty with me for that reason. Wait staff, too. (I still eat at the nice restaurants, though less frequently solo.) Any and all awkwardness is really in your own head - being solo doesn't scratch of the surface of odd in the minds of most CMs.
And as a single-rider, it's often hilarious to chat with whatever group you're thrown with. I once ended up on RNR Coaster with a family from Jersey. I kept cracking goofy jokes, and the mom insisted that she wanted them to adopt me. I also had a really amazing dinner at one of the Wolfgang Puck restaurants - I sat at the bar, adjacent to a couple, and we ended up having a really fantastic discussion.
And, to me, that's one of the fun things about going solo - everyone around you is there to have a good time, so they tend to be more chatty and more friendly. And I'm not a particularly social person on the whole.
As a frequent solo - my biggest suggestion is to go at the more "adult" times - especially during the Epcot Food & Wine Festival (which is SO AWESOME). I went to a Star Wars Weekend this year for the first time, and it was also amazing - I ate solo at the Rebel Hanger, and there was a guy at the next table doing the same. When he left, a solo woman took his place. But there are a lot of us out there at any time. (I can say personally that I feel less self-conscious with fewer parents and really young kids around, but those types of awkward moments are generally pretty rare.)
Solo trips are really their own animal. You can really do so much - and, sometimes, it's great to call an audible and do something completely random, because why not?
And if you miss the group vibe, try one of the less-intensive tours (eg, Epcot Undiscovered or Keys to the Kingdom). You're in a bigger group with random folks, and it's always been fun for me.
But I always keep in mind that the solo trip is about me having fun, not about what I might be missing by being there solo. To the point, I can actually say that there are times that I've *prefered* a solo trip.
I'm happy to throw out random solo ideas, if anyone's interested.