Yeah that can be tricky, which of course is the main reason I recommend learning on 4-string.
I might be able to help with some brief comments though - any time I say “rest your thumb on the pickup,” rest your thumb on the B string. And when you’re resting it on the E string, make sure you angle it so it’s catching the B string as well.
Other than that, muting is a tricky beast. If you can’t cover it with just the right thumb (which is the simplest/easiest method, but doesn’t always work contextually), you’ll need to find a spare finger or piece of finger on your fretting hand. Here’s an example:
I’m fretting a G on the 3rd fret of the E string with my index finger, but my middle finger is reaching over the E string to lightly touch the B string to keep it muted.
It takes a lot of work to develop fretting hand muting habits, because they change depending on which fingers you need for fretting in a given passage of music. But you can start playing with it in little bits, just looking for a free finger to mute with any time you hear ringing you don’t want.
Does that help?