I've discovered that my brain always prioritizes my guitar playing duties over my singing duties. If the guitar part is unfamiliar to me, I always mess up the words in an embarrassing way (which explains how my brain is wired). However, over the years I've discovered that I can get a lot more gigs if I sing more and help with harmonies. With the budgets for live music being less than they were, a lot of places aren't willing to pay as much for a live band anymore. So working bands are always looking for people who can cover more things to make the money go further. One of the best things you can get, to be more valuable as a background singer, is a harmonizer pedal that works with your guitar, allowing you to create great harmonies with only your one voice.

I had to spend some time getting better and playing and singing at the same time, when I started to do more of these kinds of gigs. Here is what worked for me to improve on this:

The key thing for me was to work out the phrasing with my singing so that I could work out my right-hand-strumming pattern to go with my singing. It helps if you play the song very slowly and pay attention to where your guitar accents need to happen relative to the words you're singing. Once you are able to identify those accent points, and where they land in your lyrics, then you can more easily keep your rhythm going while singing, since you are more aware of how the two go together.

Then, practice the song slowly so that you can keep that rhythm part going on your guitar while you phrase the words you are singing. Some songs are much easier than others, to the point where you don't have to concentrate that hard. But other songs take more work if the guitar part is more syncopated and not so regular.

--Jim