I have to transpose those three flats ( E, B and A) to sharps (D,A & G), I think.
No, you don't have to transpose anything:
Depending on whether you're in a flat or a sharp key, the notes should be referred to in terms of their appropriate enharmonics.
In your case, although most chromatic layout charts say that when the slide is pushed on a D hole, the resulting note is D#, that's correct when in the key of E Major, for example.
But if you're in Eb, there is usually no reason to think of it as D# - in the key of Eb that note is Eb, not D#. That is how a player should think of it and that is how it should be written.
The slide notes should be considered like the black keys of the piano - their names change depending on the key. In E Major, the black key following D is D#, but in the key of Eb it's Eb.
Nothing has to be transposed at all, because the 'virtual' instrument layout changes based on the key being played.
Here is an arrangement of the tune My Funny Valentine for Chromatic Harmonica. As we can see, it is written in Eb, with no transposition for the chromatic:
Eb is notated as
It is found in this book:
(At first I thought the question was about playing the song yourself, but perhaps you just mean writing a the music for the chromatic, in which case the following is not relevant to you as a composer, but to a musician playing the chromatic)
Practice your scales, going through the Circle of Fifths, and focus on spelling each scale correctly using the appropriate enharmonics. After a while it will become natural to name the slide notes based on the key you are playing in.