I believe the main problem here is the lack of swing playback. In the most commonly used notation softwares, there's an expressive text which states "Swing!" or the like. This by default swings the 8th notes, which would probably help transcribing the piece a lot. It's also necessary for telling performers to swing the 8ths.
Also I believe you were asking for a reference previously, so you may use this first bit here as a reference:
(https://i.imgur.com/gXGAgUA.png)
I'd still definitely recommend transcribing this part yourself instead of copying directly from here, but it's there if you want it.