I sing mainly harmony BVs in a Jam tribute band, but take the lead on a couple of Bruce Foxton sung tunes. My approach to singing and playing is this.
1 learn the bass part ( you must be able to play it without thinking about it.... so go over it many times.
2 learn the lyrics ( I often find this to be the most challenging thing to do ) because you will trip yourself up, when playing, if you're having to think about "what comes next lyrically" ?
3 Try to play and sing both parts together. You may find certain parts are a bit tricky, especially up beats in a fairly busy bass line. So slowly play and sing those parts, and work out where the word / words fall on the up beats.... and play those sections, over and over until it kinda syncopates with the bass. (You will find , you have more time and space in the song than you originally thought)
Then go into rehearsal full of confidence......and completely f**k it up. REPEAT steps 1,2 and 3 until you nail it.
It's not easy, but with time it gets easier and faster to learn new tunes. * We'd just nailed " Beat Surrender" in rehearsal , ready to gig new tune, then the pandemic hit.....Gutted. Stay safe everyone and keep practicing.