Hi Gene,
I have learned to play in multiple keys on a 12 hole C chromatic (not as many as you by the way) with most of the material I play being in the flat keys, however, to say that I can play every song I know in every flat key would not be the case, as I play each song in a key or ( multiple keys in some cases) that fit the harmonica and sound right, neither too high or too low.
As for the sharp keys I am really only reasonably proficient in the key of G so I have an A chrome that I use for songs in A, D and E, but as I say I rarely play in those keys so I decided some time ago to not bother learning those keys on a C. However, some songs I play do modulate to the sharp keys and I can generally get though, so I am not completely helpless.
I also have a Bb chrome that I use occasionally for playing in Bb and Eb mainly for easier playing more difficult pieces but it's not very often. While I am primarily an ear player I do read music, but I'm am not proficient enough to sight read a song I don't know, but I do use sheet music as an aid to learning the more difficult phrases in a song. Also I can only read music playing a C chrome, I'm not skilled enough to transpose on the fly. However, if I do use sheet music to learn on a Bb of A chrome I do the following. For example, if I want to play a song in Eb on my Bb chrome I use sheet music in F and pretend I'm playing on a C chrome, I find this works well for me.
These are the keys I can play in on a C chrome follow by the corresponding keys on a Bb and an A.
Chromes
C Db Eb F G Ab Bb
Bb B Db Eb F Gb Ab
A Bb C D E F G
I am a believer in what works best for each individual player, the chromatic harmonica is not an easy instrument to learn, and extremely difficult to master, if one can learn to be reasonably proficient on this instrument it's a significant accomplishment.
Rich