If you’ve ever played drums before, the same independence you use to play those is the same that goes into playing and singing at the same time. Back in the early 00’s I was in a prog-metal band and the singer quit before we had some gigs so I had to school myself on how to play that stuff while singing, it was a learning experience for sure.
Whenever I got stuck, I would slow the section down in my head, figure out what part you’re doing first, hitting a chord/note on the guitar and then singing or vice versa, I’d have to play it in my head before I could physically do it, then slowly go through the motions and build up to actually performing it. After a while autopilot will kick in and you can focus on one part or the other a little better. A lot of times I’ll mute the strings with my right hand and just practice singing over muted strums, it takes one variable out if you’re concerning yourself with chord changes. Your fingering hand can generally do the autopilot thing on it’s own once you’re strumming/picking hand and voice are doing it’s thing.
The initial phrase is generally the hardest part, once you can get it started the rest tends to come out a little easier.