Just to clarify - make sure you're naming the strings correctly. The top string (1st string) is the one nearest the floor and the bottom string (6th string) is the one nearest your chin.
So the string you play at fret 2 is the 5th string. It's not ok to touch other adjacent open strings as that will mute the sound of them. It's not an easy chord and will take some time to get it right. (It took me months to get it right
)
There was a thread posted recently talking about different fingerings you can use to play G - all of them are correct and you may find one of the other ways easier.