It only works when you stop trying.
I'm serious.
Do this, play a simple three chord progression. Like G, C, D. Chords you absolutely don't have to think about playing in that order. You just strum. The ones that are most natural to you to play. Then sing some stupid crap over the melody that comes to your head. It's important to sing stupid crap so that you enjoy it, and also coz it helps ease your anxiety about your Sing & Play combo performance. Once you do this, you'll get a feel for what it's like to play AND sing, once this feeling is absorbed into your system, you'll find it a lot easier to sing and play whatever songs you want.
About the stop trying bit;
If you can, remember how it was when you first started playing fingerpicking? You are picking some arpeggio, and are using your thumb to pick the bassline, and the other fingers to pick the other strings. That PIMA thing. Now, if you wanted the thumb to pick the bass string ON THE START OF THE NEW MEASURE, at first it was hard, but when you stop thinking about it, that's when you get to do it. That's my point.