A prerequisite to Transposing Harmony like a musician is to conceive and hear the music in functional musical terms, not just as an arbitrary sequence of dots.
Consider, for example, the standard turnaround chord progression CM7-Am7-Dm7-G7.
The key to success is to understand this chord sequence in functional terms… as IM7 – vi7 – ii7 – V7.
Then, if you want to play it in another key, all you have to do is think and play “IM7-vi7-ii7-V7″ in the new key. In other words, you do not “transpose” at all. You simply play those functional chord changes in the new key! No brute force calculations or mental gymnastics required! But what IS required is the deep understanding that comes from mastering scales, chords, and chord progressions… as meaningful patterns with your intellect and ears fully engaged!