Try fingering a G like this: X 10 9 0 8 10 (it's like a C shape with the pinky added to the first string).

Try a D chord like this: X 0 0 7 7 5 or 5 0 0 7 7 5 or X 0 0 11 10 10

Try A like this: X 0 7 6 5 0

All these (and more) allow access to notes around them which may be melody notes.

A neat trick is to get or make a fretboard map with all the notes indicated (PM me and I can send you a very good one) and highlight only those notes contained in the chord you're looking for. For example, for a G, highlight alll the G, B, and D notes all over the fretboard. Suddenly you'll be faced with dozens of possibilities for chord fingerings. This is a technique I use regularly to open the fingerboard to the fret hesitant student.

Good luck.

Peace, Mooh.