I just posted a few minutes ago asking for opinions on Ada. Its slightly more - what's the word - I almost feel like its a hippie name compared to Ava, which seems more sophisticated, but it's along the same phonetic lines.
Personally, though, I would just go with Emery. I actually like that name a lot - it's really beautiful. Don't worry about correcting people. You might have to correct 2 people out of 100, and those two people will be easily and politely corrected, and then life will move on. Not a big deal. It's not sacrificing a name you both love because you're afraid of correcting a few people.