I really cant seem to figure out how to mute with my right hand. Ive been doing it by resting my palm on the strings and tried to dampen lower by moving it to wherever im playing. For instance if I was playing something on the D string then moving to say a bend on the high E my hand would slide down to make sure the B,G,D and A were muted (my hands are kind of small so a lot of the times I dont have the low E muted which is another problem). However when Im playing with lots of distortion it makes some noise as Im making these shifts with my right hand which in turn makes my playing sound unclean

Im just really confused cause I watch this guy play and he doesnt seem to be muting the lower strings at all with his right hand and his playing is like perfect

I actually had problems with muting before, and I would suggest that you play some really heavy music.. like, dunno.. Killswitch engage or lamb of god kinda stuff (even though you're into Satriani). Because when you play this kind of music you will indirect force yourself to mute, and I believe thats the way to go... I learned to mute properly this way It's just a suggestion though.
String damping. You see those little X's in between chords in tabs? Well, you mute those with your left hand. Continue to mute with your palm, but discipline yourself to also mute with your left hand while not playing.
on some equipment it's just harder to mute.. like my peavey XXX combo with my tubescreamer. it just seems to focus on noise rather than the notes i'm trying to play. however things are much cleaner when i play on my little 15 watt practice amp or an amp simulator on the computer.
Muting is a VERY difficult subject to tackle, since many different people employ many different methods to achieve mastery over it.

I personally use the side of my thumb on my picking hand to mute the strings, alongside a solid fretting hand muting technique which is best explained here.

Freepower is also an advocate of the thumb muting technique that I mentioned above.

Remember, there is no substitute for practice. It will frustrate the hell out of you at the beginning, but just keep grinding away. You WILL get it.
I mainly use the fleshy part of my palm but also use other parts of the picking hand depending. This takes care of the lower strings, but it must be mobile(moving up and down accordingly). My index finger on the fretting hand takes care of the strings higher then the one i am playing(as the underside of the finger mutes those) and also one string LOWER then the one i am playing(with the tip of the index touching it slightly). I may also mute with my other fretting fingers if they are unused but i really have to work on making that second nature. Some people(satriani) keep noise down by using the unused fingers on the picking hand too(especially during/after legato).
Yeah you can mute with your fretting hand by using the fingers you arent using ,for example some players keep their first finger free and have it barring the strings behind the notes your playing to mute them. If its making too much noise ,try having your hand already muting most of the strings you know you arent hitting then just raise and lower it accordingly