I either sing lead vox or am the primary backing vocalist in most of the bands I work with.
I can tell you thst developing your vocals will land you a lot more gigs than just playing. Most of the successful bassists I know have put just as much time into developing their vocals, and being able to harmonize with the lead vocalist. Finding thirds, fifths, sevenths, that will get you gigs, trust me. Bass is a backing instrument. Develop your voice as a backing instrument as well and be able to offer two instruments for the price of one.
Singing and playing bass is awkward. At first. But so was playing. The more time you put into this the better you will get. Building stamina in your voice is paramount to singing well, and fluidly. At this point I can separate my vocals and bass lines and sing/play independently of each other. That took serious time to develop, but well worth it in job opportunities.
My suggestion is to find a song that isn't that taxing on your voice. One with an easy bass line that either chugs along or follows the vocal line rhythmically. Practice the song, and whisper the words over the bass line to get the flow of the vocals down. At first you will pay more attention to your voice than bass line, hence the easy chugging bass. Less to think about.
As you develope your new skill, pick songs that are more challenging in both areas, until it becomes second nature. This is the easiest doing backing vocals. Work up solid backing vocals, staying with the harmony (not the melody) so you can separate them. Harmonies are nothing more than simple vocal chords.. (I'm sorry, bad pun).
This is the best kept secret in a gigging musicians arsenal. Develop it and prepare to get the gigs.