I am learning to self-accompany on harp, and I have experienced exactly the same problem.
You've already got some good suggestions. To add to them: something I've found beneficial is to conceptualize the thing-that-hands-are-doing and the thing-the-voice-is-doing as a single thing, and practice accordingly.
To explain that -- one approach already discussed is to try to become automatic on one or both of the parts (voice or instrument) and hope that carries one through. That's not wrong, but I think it's complementary to the opposite approach of learning the two parts together. When one studies the two parts separately and attempts to get them so automatic they can unspool from one's hands/voice like from a tape, one never figures out the persnickety details of how the two parts fit together. But it's the persnickety details where one get derailed.
For instance -- btw, the example I have in the back of my head while talking about this is Dowland's "Come Again", if it that illuminates my examples -- if in learning the instrumental part, I hadn't noticed that I was not quite playing the 16th note runs in strict time, I may think that I have the instrumental accompaniment down, but when I try to put it on "automatic" and sing over it, every time I hit one of those bars, my voice part is going to get to the end of the bar first, and I'll be all lost and discombobulated, thinking "Wait!? What just happened?!" There are two solutions: One is to pull out the metronome and discover those 16th notes are lagging, and get them up to tempo by drilling them. But what if I want to linger over the 16th notes? What if, say, I am milking the dirty joke in the lyrics or just generally getting my artiste on with some rubato? Then what I want to have happen is for the voice to be rubato in sync with the instrumental part. And the only way to possibly get that to work is to have a very clear and precise understanding of how the rhythms of the vocal part sync with the rhythms of the instrumental part.
To that end, one can, e.g. make sure one has the count for the joint piece -- hands and throat -- right, and then work on it reeeeeeeeaaaaaaalllllyyyyyyy sssssssssllllllloooooowwwwwwllllllllllyyyyyyyyyy until one reliably gets it precisely right. Then work on cranking the tempo back up to where it needs to be.
Now, this approach may make more sense for classical guitar, but if you're doing anything at all complex with picking, it might pertain. I decided I wanted to be able to self-accompany on Sumer is icumen in, which has "pes", a ridiculously simple repeating bass riff -- but for me, I found doing it required that I actually figure out how every individual note of the melody lined up (or not) against the corresponding note of the bass part.