The best advice I can give is the following:

If you're playing righty, have the guitar on your left leg
elevate the head of the guitar a bit
keep your thumb in the middle of the back of the neck
play on the very tips of your fingers

You should find with a combination of these you can find a position where you get more arch in your fingers over the strings, and hopefully less muting of the adjacent ones.

You can also try playing chords differently. E.G:

A major

e--0
b--2 <-- ring finger
g--2 <-- middle finger
d--2 <-- index finger
a--0
e--x

or:

e--0
b--2 <-- ring finger
g--2 <-- index finger
d--2 <-- middle finger
a--0
e--x

Hope that helps a bit. Pay close attention to what you're doing when you play and how you can influence it. The solution is there, you've just got to find it. And yeah, I have big shovel hands too :)