Hi Ann! Yes, if you are tuned a little low (CGC) then the strings will tend to vibrate in a wider arc when picked or plucked. This would explain why you get the buzz only when picking the string a second time, while it's still vibrating from the previous note. Three things can help this-
1) try tuning up one step to DAd and see if it is improved by having more tension in the strings... or,
2) stay in CGC and put on a set of strings that are slightly bigger gauge (instead of a .010 for example, try a .012 or so- this will also add a little tension and decrease the vibration arc of the strings... and
3) Ken mentioned something about this- try picking with only the very tip of your fingerpick and not the part nearer your fingerpad. It sounds like the already-vibrating string is prematurely hitting the pick right before you actually pick your note. Playing closer to the pick tip end would help this.
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