From Ionian to Mixolydian

I'm trying out Mixolydian (going from CGG to CGC).  However, this of course causes a flat note at the 6th fret.  I don't have a 6+ fret.  I know the scale starts at the open melody string, instead of the 3rd fret, but how do I play the same songs in Mixolydian as I could in Ionian?  Or don't I? I should mention that at this point I only play with a noter.  Thanks