To Anna's credit, she didn't end up stumbling over her words quite as much as she had back at Elsa's coronation. That would've been... awkward. Not that awkward wasn't Anna's middle name or anything.
"You're beautiful too! Probably more than me, because, you know, you're... the Queen and stuff." That could've come out better! "I don't want to be without you ever again, Elsa. I'll stay with you."