So from most videos I am seeing, it seems that classical guitar technique requires you to hold your hand away from the strings.
If I pluck an e on the 2nd fret of the D string, and then stop that e, i still get a lot of noise from my low E string, that started resonating while the higher e was playing.
To me, this sounds very muddy. I haven't seen anybody on YouTube doing any right hand-contortions to avoid this problem, nor even mentioning it.
What am I doing wrong?
This is part of the classical technique. The resonances of the other open strings help produce a full sound, see Carcassi's text. It is supposed to be there. The strings that resonate will have the plucked note as a natural harmonic so it should not be dissonant. However you do need to gently mute when moving from one chord to another and you will get some conflicting dissonant harmonies. Depending on the situation you can turn the meaty part of the thumb down towards the strings when switching from one chord to another, the outer part of the hand near the pinky works too. Relaxing the left hand and flattening it over the strings is a possibility.
It takes time to develop these techniques in such a way that they are graceful and don't cause other problems like unwanted squeaks and scraping noise.
First, it sounds to me like you are not aware of proper technique. Get the idea out of your head that the open resonances need to be stopped. In general they should be encouraged.
