[New 12 String Mike]

Valerie is right, you don't mute the E string when playing the A chord that way.

A modified way would be to barre the finger as Gord's doing in the photo and tuck your second finger in on the B string. That can help you from muting the high E. When you can play it with just the single finger you're OK. When you can do it on a 12-string, consider the chord "conquered."

Jesse, on the index free G you can play the 1st and 2nd finger positions of a C chord while keeping your bass G and or your high G. Your 4th finger can also slip under your 3rd finger on the bass G to add the standard bass C of the C chord. IYCRMM uses that chord and as I'm sure Cathy can tell you its how you play Travis style on Simon and Garfunkel's The Boxer.