In my experience with the guitar, I think I do best with strumming. But it's always difficult for me when I have try to keep singing while strumming. How can I learn to do both?
@Mark is right on. You've gotta practice enough so you can do it without thinking, even without the instrument.
My trick for getting there… play and read the news at the same time. It takes practice; at first you'll only be able to read a headline and then you'll mess up a chord, but eventually you can digest entire news stories while you're playing through a song. That way you don't have to actively think about DOING something while your muscles are going through the strumming and chords to a song.
I've heard of people who play in pit orchestras for broadway, etc., being able to read books while they play. Could be a rumor… but if you play eight shows a week for a year, I bet you get close to that!