The Em11 and A+7 are II, V in D, the E sharp (F) in the A augmented would suggest D minor, but the Em11 is D major, so you could add a bit of interest by swinging between those two scales, same with the Dm11 and Gaug7 - either C major or C minor. Ultimately it's down to your ears, but yes, D and C are good places to start with those particular chords.
As for the Aaug7 and Bbm7 you could look for common chord tones -
A+7 being A, C sharp, F and G
Bb-7 being Bb, D flat F and Ab
Since Db and C sharp are the same note, and the Fs are the same then you could use those as pivot points in between the chromatic movement.
Or you could indeed go for a modal approach.
Hope this helps, 'Speak No Evil' isn't a tune that I'm overly familiar with, so this is all just theoretical. Let me know how you get on...