I don't know if this has been addressed yet, but
Em names are VERY common.
Emily and
Emma ranked #1 and #2 respectively last year according to the SSA. Calling her
Emmy or
Emma wouldn't distinguish her in any way. In addition,
Grace is also very common both as a fn (#20) and as a mn. Still, it's not as common as
Em names. Have you considered
Grace as the fn? If I had to choose between
Addison and
Emerson, I'd have to go with
Addison since
Addie isn't as common as
Emmy and
Emma.