My friend is having the same issue. I basically told him the same thing Frederic suggested. However, I didn't tell him about starting off a capella. When I was thinking of the method, I was only thinking of the playing, so I never considered the a capella beginning. That's actually really clever, and I completely support Frederic's suggestion.
However, that's not how I learned.
I've never played guitar until June 2009. That's when I got my first guitar. I learned how to play chords, and some lead. Every time I tried to sing and play, I would just get lost . . . very, very lost
Some time around May, I bought the Beatles Rock Band. I have been a passionate fan of music games since DDR. Around the time I got BRB (haha), I owned GHWT, GH5, GHVH, and GHIII (in that order). I've beaten all those songs on expert guitar, and all the songs on GHII and Rock Band when I went over friends' house who had them. So, hand coordination isn't an issue for me. But the reason why I bring these games up is that I don't encourage you to try to sing and play on any of them (that include singing). I suggest sticking with BRB. That was my first try of playing and singing, and it went pretty smoothly. I started with expert guitar and medium vocals (sorry, no prior singing background). Within a week I moved up to expert vocals, but went back to hard because the grading is very harsh to the point were singing becomes a distraction, and replaying a failed song a few times because minor mistakes isn't worth the time. Another reason why I suggest BRB is because of the songs. With regular Rock Band/2 or GH5, there's variety of vocal styles, guitar styles, causing inconsistencies with note charts that can be sung along with and difficulties. BRB is pretty consistent because anyone who is doing lead vocals is providing some sort of guitar instrument simultaneously, making it realistically convenient. Also, The Beatles' music is easier to grasp after a few listens. Really catchy tunes, enjoyable to listen to, and the guitars are memorable. Lastly, the display of the singing graph and the guitar note chart are a lot easier to follow simultaneously. I actually prefer GH's note chart, but for singing and playing purposes, RB is more comfortable. I'm still not at the level to do full on rhythm guitar with lead thrown in. At least, not the lead stuff that is obviously lead material. I can do embellishments and melodic lines, but anything that stretches too much in lead territory takes me more than 10 minutes to get down. An example:
I can do what the acoustic is doing here -
https://www.youtube.com/watch?v=MxFdllo-1qs
and sing at the same time. Took me no more than a minute to really get it down.
Or do something a little more along lead, like here-
https://www.youtube.com/watch?v=48VUJOCNSc8
and sing at the same time
But I'm sure I can't sing and play Here Comes the Sun as fast as I could learn the above two examples.
You might say, "Oh, that's a video game, blah blah blah", but this is a fun, easy, and proven way that builds the coordination. I mean, how else do I explain I was able to play A Hard Day's Night singing and playing almost effortlessly? Well, I was already getting the hang of that song before purchasing BRB, but I'm sure it was BRB that made it possible to do it with ease, as well as just picking up any song for the first time and easily sing and play at the same time.
If you can't shell out the cash, than I say try Frederic's suggestion, which I think is quite possibly more effective.