Many great tips in this thread.
I've been singing and playing bass since the late '60s. Something recent threw me back to a very important point. I started giving bass lessons to someone who never picked up an instrument before. When you first picked up a guitar or bass, remember what a klutz you were..making your fingers move across the frets and strings was impossible, never mind finger picking.
To add singing to playing is just like starting all over. Practice, Practice, Experience, Experience.
Learn your bass part backwards and forwards. Learn the lyrics same way. Dilute your bass part to as much as you can do and carry the vocal and keep some kind of rhythm. As has been noted here a number of times,"the folks listening will never notice the missing bass notes, the vocals are what they are listening to. The more you do it, the easier it will be.
Major point!! Guitar players are generally playing on the rhythm, bass players are playing something else which crosses the rhythm from time to time. That's why guitar players, rhythm guitarists are often the lead singers. Lead guitarists know exactly what I'm talking about and will sympathize/help you to accomplish your chore.
Practice with tapes, metronome, drum machine or your drummer. It does get easier...eventually.
Someone a while back, mentioned "disturbing the neighbors". My band rehearses direct into the board, then to a headphone amp, as well as with digital drums. The only thing heard in the room, is the vocals. Side benefit, you can adjust your headphone volume.