Christopher, I hear ya... Try playing these measures (note the muting required), I had a terrible time with muting. This is a D02 piece (wound up being the D02 Exam Piece, so I felt I had to deal with the problem). I don't know if you all have problems with this (string muting) but I sure did.
I found the muting of certain notes of this piece difficult (buzzing strings, etc.) but never realized my thumb was in the way until I made close-ups of my right hand. My thumb was hitting my index finger which interfered with muting. Still a work in progress as you can see, muting is still hit and miss to this day, with difficult pieces (i.e., Schiarazula Marazula).
I use Polyphony-like exercises as part of my warmups, though I hated that sort of thing when I first encountered it.... Just an FYI.
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1989 Takamine C132S, Aquila 19C Strings