YMMV, of course, but here's what worked very well for me.
I started singing backup. To be able to do that, even, I figured that one or the other (singing OR playing bass) had to become "automatic". That is, I would have to be able to do one or the other without actually thinking about it. Being new to singing, I chose to learn to play my bass parts on "autopilot".
Two factors became important, to be able to do it. First, I had to "scale back" on the technical difficulty just a bit. Second, and probably most important, was that I had to learn to play everything without looking at my hands. I didn't quite realize it at the time, but that ended up being the key to everything I did, to become quite good (if I do say so myself) in a lot of different ways.
With practice, I got so that I could play more complex and difficult parts "blind" (i.e. without looking), and got to where I could sing lead as well as backup. Eventually, I got to the point where I could improvise both vocally and on the bass, then do both simultaneously.
I practice playing everything without watching my hands, and make it a point to never "cheat". After all, the frets & strings don't move, so why should I have to look? As a female friend put it so well, discussing why some women seem to have to use their mirrors to put on makeup while driving: "See this face? I've had this face for 30 years (at the time)! I know where everything is!"
When I learn something new, I try to learn my bass parts very basically first, to the point where I can go "autopilot" & start rehearsals with the band. Then, I add vocals, again very basically. As I learn both parts, I'll start improvising and embellishing both as I get comfortable. After enough practice to achieve a certain comfort level on my part, I'll be able to easily toss embellishments out at will, both vocally and instrumentally. For my purposes, that's about as good as it gets.
JM