This is what was hardest for me to adjust to on my 7-string. I have pretty big hands though, but I still had to put my right hand flatter on the strings near the bridge to get everything silent. Was a bit awkward at first, but you get used to it.
Advice? Hmmm, practise I guess. Just really pay attention to properly muting everything at all times. After a while you'll do it without thinking about it. The 'awkwardness' should go away after a while, change is always weird.
Oohh, btw, congratz on the new guitar, I'm actually thinking of picking up a DC727 myself. And I was wondering how the pickups are on it? I think their business would double if they let the customers choose their own pickups, damn them! ;)