I'm a bass player+lead vocalist/backup vocalist depending on the gig.
+1 to everything that's been said here. It took a bit of self-awareness for me to sort out my problems with singing and playing bass. I started with harmonies (which is probably harder than starting with singing lead vocals). Here's how I do it (which is pretty much what others have already said)...
I work on the bass part first until I can play it kind of mechanically without thinking about it. I noticed along time ago was that I was "singing" the bass part in my head like a melody while I was playing. Having made that observation, I understood why I couldn't sing a vocal part that was counter-melodic and often counter rhythmic to the other part I was already "singing" (the bass part). So I developed a habit/technique where I make a conscious effort to learn the bass part well enough to not have to sing it like a melody in my head in order to play it. In the mean-time, I practice the vocal part separately. When I put the two parts together I pay close attention to the placement of notes in time for both parts. It's sort of like I imagine the two parts on one staff together, at least rhythmically, if that makes any sense. Or maybe it's more like reading piano music where you have right hand and left hand things happening simultaneously. My point is that the key to doing this is knowing exactly where the notes for each part land (in time) as I count out the beats of a bar. That sounds absurdly rudimentary and understated, I know.
Now that I've been doing this for a very long time, I don't actually have to think about it any longer, but that's a basic description of what worked for me when I taught myself how to do it, and that's one way that I've used to teach students to do it. This method sometimes reveals the need to practice different rhythms, both vocally and while playing bass, and also to think more analytically about rhythms. This kind of thing is always harder for me to describe in writing than it is to just show someone in person. And then... practice.