What are some ways of learning to sing and play an instrument at the same time?

I’ve played guitar about 5–6 years, self taught.

I know the most obvious question would be, “Keep practicing!” but I have been, and I can’t maintain steady rhythm and sing at the same time (forget finger-picking, or attempting to do anything technical!).

I can play and sing to a few very basic songs (Creep – Radiohead… barely), yet for songs that require a bit of an off-tempo (Blower’s Daughter – Damien Rice) approach, I fail miserably.

I think the proper question would be: What types of tricks or thought processes do you use when tackling a new song?

For example, when I was first learning sweep picking, I imagined just that – sweeping the pick across the strings, one at a time til I built up speed and control of my left-hand roll. Do you use any of these “tricks” when teaching yourself how to sing and play (pianists, too!)

I can’t jump over that hurdle of doing two things at the same time, and I can’t stand it :(

Any help would be much appreciated.