Shy toddler/sibling relations

My 3yo daughter is lively, imaginative, and extremely talkative at home, but in the presence of non-family members she is extraordinarily shy. When we have play dates with the neighbor girl that she has known her whole life, she will play a bit but barely talk. In the presence of strangers or even people she sees once every couple of weeks she will try to hide. Ever since her third birthday, we have been bringing her to Sunday school where she is either the only child or one of two, and she continuously squirms and tries to hide her face in my lap, etc. despite my instructions to sit up straight. How should I best handle this? Apparently my husband was like this as a child, so maybe it’s just a family trait. I plan to ask my MIL what she did. I stay at home with her, and she was an only child until six months ago. I’m wondering if I ought to get her into a class, perhaps ballet. I’m so against promoting too much attachment to peers on the one hand, but she is really lacking in social skills even for her age.

That brings me to my second question. (If it would be better for me to ask this next week, let me know and I’ll do that instead.) She has had a difficult time adjusting to having a sibling. I had really tried to be an “attachment parent” a la Dr. Sears with extended breastfeeding, cosleeping, baby wearing, etc. (I’d love to hear your thoughts on this, maybe another week!) I did not ease her out of some of these things before the baby was born as I probably should have. (I couldn’t wrap my head around not her not being the baby.) She told me the other day she doesn’t like her brother. She told my husband that she loves him but doesn’t like him. This is totally understandable to me. We still spend tons of quality time together, but things have definitely changed. How can I promote good sibling relationships from the start?

Thanks for your wisdom. I’ve been listening to calls starting with current ones and working my way backwards, and there is such a wealth of wisdom in these archives.