First, you can name her Anna and kids will still make fun of it. ANY name will get made fun of--it's how kids are. Your family can get over it. My sister called her daughter Ally--I hated it. I got over it and when she got older, she dropped it for Alex (which I adore). Her other daughter's name is Sydney, which I also strongly dislike. Again, I got over it. It's not my kids, it's not my business!
Personally, I'm with you--Ann is boring, so is Anna. Annabelle is beautiful (well, since it partly MEANS beautiful... lol). I even have a character in the book I'm writing who has that name and comments that she hates that everyone calls her Anna when she wants to be called Belle (just brought up that memory, lol, that was years of writing ago at the beginning of the book! yes, I'm slow). I've named a few fictional characters Annabelle and I think it's an awesome tribute name!
As for Delilah... there's a song for most girls' names if you look for it. And no one ever seems to like the song for their name. Don't worry about it. And no, it doesn't sound like a country singer. For crap's sake, I'm in Missouri and didn't get a slight 'South'/'Hick'/'Redneck' twinge from it (and I've got quite a few hicks/rednecks in my family--I'm not one of them, they drive me nuts--and DH, too, who feels the same about the rednecks in his own family).
This is why we're not sharing our baby name. It's not their baby to name and they will get over it or create nicknames that they prefer.
I have one friend who wanted her son called Xander. Well, she was going to name him Alexander, but her mother or MIL, can't remember which, said she didn't like Xander and would call him Alex. Well, my friend didn't like Alex. So her son was born and she named him just Xander. Not to spite anyone, but because that was the name she wanted him to have.
Use the name your DH and you love. Tell your family that you're saddened that they don't love the name like you do, but that it isn't their child to name and you hope that they can find some good in it. Tell them to call her Anna if they're so inclined. She'll choose her own path, as far as names go, as she grows.
My daughter is Lillyanna--she has a few choices for nicknames in there (Lilly, which we call her, Lia, Lianna, Anna, Ann) and that was deliberate. It's my favorite name and DH thought it was pretty, so we went with it. I can't imagine anyone not loving Annabelle! I hate that I never had an option for nicknames with my name. I believe in giving as much option for that as possible