As someone who has his own social anxiety hang ups, I can understand where you're coming from. But actually, this game is fairly friendly to that kind of mentality! Especially if you're goal is to enjoy the story.
While there are some flashpoints that carry out a good bit of story, a lot of the plot seems to play out through your class storyline or the planetary storylines. The good news is that for these, you can experience the entire story solo! Unlike a few other games I can think off where if you want to finish the storyline in an area, you'd have to find a group and do a few group quests (that always drove me nuts).
I haven't run into any heroic quests (2 or 4 person) that are absolutely necessary to understanding the plot. Oh they'll flesh it out a bit more maybe, but it's not like the big conclusion to the planet's story arc will make you group up. It's one of the things I've really enjoyed about this game so far.
Sadly, at the moment at least, when you finish all your storylines and hit max level, there's not exactly a ton of options beyond grouping. Dailies, space missions, running in circles around the fleet and admiring everyone else's armor? But until they add more max level solo content (if they do? I think I heard someone around mention it in the pipelines but it wasn't in yellow, so I don't know if there's anything official on it) you can also explore one of the other 7 class storylines, experience the game from the other faction's side, and level up another character.