rushfan....i started out using my thumb as an unmovable anchor like yourself and was quite pleased with the results... until i started recording in studios were the emphasis is on a "precise ,clean as you can sound"...then i had to go away and look at how i would achieve this..my left hand muting was good but the problems occured when playing the g str for any given time as the other strings set off in sympathy.try it and tell how do you achieve this ?

mazda, keep using the moving technique i wish i had started that way..but it doesnt take much to relearn..one point i will make is try the thumb when playing the e string against the front of the back pick-up instead of on top..i found this imitates the same position as for the other strings and also feels smoother when moving to the a string...
as jaco put it "the hardest thing is making the instrument not make noise"
cheers markj

...dont forget they say "cleanliness is next to godliness"