Fortunately the other two guys both sing, and I've been singing harmonies in the shower for many years, so I decided to help out and sing a few and get better at background and harmonies. Harmonies seemed easy; heck, someone else has to remember the words, so they control the vocal timing, but hitting the right note wasn't always easy. We pull off 3-part harmony regularly, like in almost half our songs.
So for backgrounds and leads at 62 I took some singing lessons; why not?! Now I also sing lead for 5 or 6 songs a night. Whoda thunkit?!
What really helped was a bit of advice from my music teacher.
First, harmony is singing the song just like the lead is, but different than yours, and the lead already knows exactly what each note is that they are singing; do the exact same thing for the harmony part, winging it doesn't work.
Second, for lead, the bass is playing notes. Singing is singing notes. Notes is notes; you have to know what notes you are singing just like you have to know what notes to play on the fretboard. Go through the song syllable by syllable and know what your notes are exactly, and then where each one fits into the beat.
And you already know where the bass fits into the beat. Like "OK, the lyric word "Are" starts an eighth note AFTER I play the C; the C is on One, and "Are" is delayed until the And." OR on a long "Baaaaaby", the bass goes from C to A on the half beat after "Baaaaby" starts. After a while it's just like you're playing both your bass and your voice, not any different really than any grace notes or adding a 5th or an octave while holding a note, and it becomes a new rhythm. This was a big help to begin with.
Next: Sing the vowels, the consonants will just happen on their own. Huge benefit knowing that tidbit!
Lastly, the tough part for me is where the singing begins BEFORE the One count; it actually starts at the end of the previous measure, like during the 12th bar in a blues progression, on the four or perhaps the three and a half or whatever. As you all know, the bass also does this a lot, like at the beginning of the song (that little roll down at the beginning of Long Train Running at the end of the guitar intro, for example,) so you already know how to do that, but using your mouth instead of your fingers makes it harder at first, but it's the same thing! Play/Sing it as a part of the 12th bar, not the oncoming 1st. Count it off at the previous One if you have to.
Anyway, I'm still not nearly there as I'd like to be, but using some of these techniques and all the other suggestions I see here, I know it's pretty good now, and will be getting better.
Oh, and then comes confidence; once that starts, it's a really cool feeling and you make a big leap.