software

My Hwy1 Tele makes unwanted rattling/buzzing sounds when I play it, even when it isn't amplified. Therefore, I guess it's the guitar (I hope sincerely that it isn't the amp, which is an old Vox AC30TB).

The sound doesn't appear when i play open strings, so it may be the clash between the strings and the frets. Do I have to refret it (the guitar is only a year or two old)? Restringing didn't work. I have a fairly high pick attack; can I adjust the action?

it could be fret wear or warped neck
check to see the fret before the string that buzz, see if there is a depression on the fret compared to the others.
Or take a ruler and place or vertically on the fret, see if there are no gaps inbetween the ruler and the fret, if there are, that means your neck needs straightening, which may cuz the buzz