Your son sounds a lot like my DS6. He still prefers to be at home with only us (he also has been telling us that since age 3 if not earlier), and he still wants me to stay with him at birthday parties. His anxiety didn't change much throughout preschool but it has gotten better in kindergarten. He had a hard time transitioning to full-day school, and his "school self" is very different from his "home self" as best as I can tell (he says little about school at home), and we think he will always have social anxiety (which also runs in our family). But it is clear when I see him with his K classmates that he is very comfortable with them and has matured a great deal socially this year. He will actually leave my side at a playground nowadays, even if unfamiliar children are there... and definitely no problem if his classmates are there.
So there is hope. DS6 has no particular diagnosis other than being HG/PG/whatever you want to call it. His preschool teachers wondered about sensory integration problems but his ped didn't think so; yet now that we have a second child we can see a big difference in their reactions to stimuli, so maybe he does have a touch of that. Still... he was very like your son at 3 and is now handling "big school" well, so we can hope you will see the same with your DS.
Oh -- and yes, definitely he is an introvert; that is no surprise at all though. We wouldn't know what to do with an extrovert around here. ha.
