Some excellent advice in here so far.
I find myself in an interesting situation these days.
I've been singing & playing guitar for 20 + years. I don't have to think about playing guitar, switching effects & singing at the same time except in the rarest of circumstances & I've been singing lead in most situations. I can play drums & sing no problem. Bass I've been playing seriously for the last ten years but mostly as a session player so no vocals involved until........
A 5 piece cover band in which I play guitar, sing a lot of harmony, & lead on maybe 7 or 8 songs started also playing as a trio 4-5 months ago to.
This is an effort to still make money at rooms not looking to spend enough for the 5 piece. The catch is, I play bass in the trio, not guitar. The trio is gigging about 6 times a month now.
All of a sudden songs that I absolutely own & can rock the crap out of in my sleep were a whole different animal. It doesn't help matters that my vocal style is very behind the beat where the song allows it.
My process for sorting this out is similar in some ways to what a few others here have expressed.
1st, I learn the bass part so well I can do it 6 beers in with out thinking about, or looking @ the instrument. Same goes for the lyrics & melody.
Then I do not think about the notes or pitches of either part at all.
Only the rhythm. My foot & the drums are the metronome & the bass & vocal are two rhythmic bouncing balls like in a cartoon & I visualize how they move in relation to one another.. Mind is completely blank other than that. Eyes closed. No thought, just feel. Again, because it's important, think only about the rhythm.
How do the 2 parts fall across the beat in a given measure? Feel where they line up, where they syncopate etc. Figure out where the trouble spots are & specifically work on those slowly until you really do understand where they land because you can't get there until where you're going.
Then, give it a break. Come back later & give muscle memory a chance to do it's thing.
Some bits will be hard, some easy & some damn near impossible. One of hardest ones for me was the easiest damn bass line.
Doing the rap parts in "Billionaire" where the rap vocal rhythm constantly evolves while just playing 1 ee 2 ee 3 ee 4 ee. on bass.
Reggae feels messed me up big time because as a guitarist while singing, I've been playing only on the up beats my entire life & now I have to do the exact opposite.
Fun stuff. I'm working on Disco Inferno right now.