Yeah, I guess the OP itself is kind of the opposite of our usual "I have to play and sing at the same time to get the feel right" crowd. I didn't get the idea that he has a problem with the singing or the guitaring on their own, but that it's the playing them together that is the struggle. I can still see where hearing the melody might help. In fact, you might just actually do the opposite of typical - record the guitar part, then sing over the top of it, then practice playing and singing while listening to that.
Part of it for me, though, has always been about adjusting both the guitar part and the vocal so that I can actually pull it off. That means sometimes some compromise. Maybe you simplify the guitar part some while you're singing. Maybe you change the rhythm of phrasing of the vocal to match or work with your strumming a bit more easily. Once you figure out how you do that, you're most of the way there. Often these things actually improve the song/arrangement.
But then that's the trap. Now you get to where the guitar and vocals are intimately tied both dynamically and rhythmically and you just "can't" do either one on its own and have it work right. I have that issue pretty often, but I have enough experience to know that I actually can do them separately - and it's usually better than when I do them together - as long as I have a scratch of both together to start from.
Anyway, like I said, you kind of just need to dive in and do it. Pick some songs you know separately, and sit down and do them together. Listen to the song on the record (or YouTube) and play along, but don't worry so much about doing it exactly like the original. Figure out how to make it work for you. In most cases it's much better to deviate from an original with confidence and style than to try to get it exact and fail.