Help please. String vibrations (dumb topic name I know)

First post on the site and it has to be a stupid question.

For some context, I'm taking lessons from a guy that's been around for a long, long time. And the lessons are moving forward at a really fast pace.

As 'homework' for this week, I'm to learn Day Tripper by the Beatles and the theory behind it. The biggest problem I'm having is, simply put, sounding like absolute shit when I actually play through an amp. I get the notes correct and on time, but I'm finding that the E and A strings seem to continue vibrating throughout the opening riff, to the point where it kind of sounds like this (https://www.youtube.com/watch?v=Ah-y6Xfk2Tg @1:20) and basically ruins any sense of progression I may have had. I only have this problem when I'm playing it at speed though, and not when I'm slowing it down. My finger pressure, movements, and everything are the same for both hands, barring the actual speed at which I'm playing. I've tried using my ring finger on my right hand to mute the E string after the first three notes, but it seems to not help, as I'm noticing that I need to keep my hand 'light.' (I don't know any other way to put it. Real technical term I know >_<)

In a nutshell, it feels like I'm in over my head because I have no idea how to mute the strings properly.


PS: If this happens to be in the wrong area or something like that please let me know.

well, as i learned from dman, its better to fret the string right behind the intended fret. if your in the middle of the fret space (space between frets) it can cause fret buzz, which is your problem i think. when you play it slow your more careful where you fret the string, but when you play normal speed you dont think about finger placement all that much. So, my advice would be to practice is slowly and gradually build up speed

Work on your mutes. Start slowly, carefully muting the strings, then GRADUALLY speed up, paying attention to muting still. You'll be able to do it properly after some practice.

Thank you for the input, TLS. I guess I'm trying to move forward too quickly. I've got the song down for the most part, missing a couple switches here and there, but for the most part it's coming together.

[edit] I just ended up thumbing the E string and fat fingering to mute the A/D strings.