Is This Sibling Name Pair Unbearable?
I do think it's a problem. A cultural connotation that links a pair of names is hard to shake, even decades after the fact. "Amos 'n' Andy" went off the air in 1960—likely well before you were born—yet you know about it, and it comes to mind right away when you hear the similar-sounding combo of Amos and Annie. Each name is just fine on its own, but put them together and you have an issue.
Yes, you could name your daughter Anna or Annika or Anneliese and call her Annie at home to try to avoid the association, but I think it would be difficult. Perhaps you'd like one of those names, or another Ann relative, on its own, although you would have to be diligent about avoiding Annie as a nickname. And that's tricky, especially once your daughter is old enough to have her own opinion on the matter.
Instead, I suggest you choose another name that can't be shortened to or confused with Annie or Andy. That's the problem with second-born and later children. Sometimes your firstborn's name just knocks another otherwise lovely name out of the running. All you can do is mourn it a bit and move on.