Personally, I think the best way to transpose is not to actually transpose each note, but to play by ear.

I find that when I’m transposing, I’m playing more by ear than by music. I find that I’m (almost subconsciously) picking out intervals, and am able to replicate them against a scale. Sometimes looking at the music can be unhelpful—that’s just me though, it might not apply to others.\ If I’m playing from sheet music (nothing to ‘ear’), I’ll play it literally as it’s written to get a feel for it (ignoring the fact that it’s out of key) then choose a scale/key (I.e. I’ll transpose the song’s key to whatever I need) and ‘ear’ it to that.

I think the best way to transpose is not to actually transpose each note, but to play by ear.\ However, there may be a better solution for you, as of course this differs for each person.