How do I eliminate unwanted string noise?

[Superman]

I am a guitarist who is a rookie gypsy player converting from rock and blues to gypsy jazz style. I am trying to learn rhythms and solos and making some progress. The particular songs I am learning are "I'll see you in my dreams", "Tears" and "Minor Swing". I am using the "rest stroke" and am noticing the volume is stronger. I am also minimizing the use of up strokes unless changing directions on an arpeggio, etc.

I am playing on a Martin accoustic guitar. There one area of concern I never hear any talk of among the gypsy jazz guitarists. MY QUESTION: HOW DO I ELIMINATE UNWANTED STRING NOISE ON STRINGS I AM NOT PLAYING? It seems that the Martin's strings are very sensitive and any touch whatsoever ignites string noise (harmonics, etc.) The gypsy style has the right hand far off the bridge so you can't mute with it. HOW DO I MAKE IT SOUND CLEANER?

Please advise,

Paul from the US

[G7+]

I do it with my left hand. I shot a video on the attachment *. If you look close, you can see that when I am leaving a string with my left hand I am also muting it. The finger rolls to the next string muting the string being left.
But I do not remember myself practicising this. I just aimed for a clean sound. This is my way, not sure how other players are doing it.
(I am left handed but I write this pretending that I am right handed)

* could not upload it so I zipped it.

[Superman]

Hello G7! Thank you for the feedback and the attached clip. This is very helpful. I do mute other strings with my left hand also. But I am going to try make sure to do it when leaving strings more. I really appreciate your feedback. Happy Playing!
Paul

[G7+]

Hello G7! Thank you for the feedback and the attached clip. This is very helpful. I do mute other strings with my left hand also. But I am going to try make sure to do it when leaving strings more. I really appreciate your feedback. Happy Playing!
Paul

You're welcome.