I'm English but I don't think the difference you've mentioned is a US/UK difference. Listen to James Taylor (a US singer) sing "Carolina in my mind" https://www.youtube.com/watch?v=Gic6B-B6rpg He says /ai/ which is how I would say it too. I think Carolina is also a Spanish name in which case you would use a /I/ to say it in the correct original form. Probably some people with the name "Carolina" choose one and some the other. You'd have to ask them how to say it.

It's Carol(ai)na. I don't think they say it differently in the UK. They do say "aunt" differently though.

I watched an interview with Carolina Wydra(House MD) She said that her name sounds with "e". Like caroleeeena. I suppose it is always up to person who has that name.