One of my very favorite Dead songs to play too. A few quick thoughts...
1. For the purpose of analysis, it could be easiest to say that the intro, chorus, and solos are in Am, the verse is in C major, and the coda is in A major. There's probably no "right" answer here, just whatever is easiest for you to get your head around.
2. IMO, to say the Am portion mixes all 3 minor scales is too complicated of a way of looking at it...really just Am triad with the bass note descending chromatically (in half steps...get it? ). A pretty common move in jazz etc. (Stairway, The Landlady by Phish, ). I suppose, in a sense, you could say you're using A natural minor > A melodic minor > A dorian, but to me it's easier to just think A > G# > G > F# within the context of A minor.
3. A lot of the chords in the progression are acting as "secondary dominants." In short, you can precede any chord in the progression by that chord's own V chord. That's why the B7 precedes the E (even though it doesn't appear to "belong" in C/Am). And the D7 (or D9) precedes the G, and the E7 precedes the Am. That D7 actually gets referred to as "V of V" (the V being the G chord in the key of C). And the B7 would be the V of V in Am (it is the V of E7 which is the V of Am).
Secondary dominants happen left and right in GD music (Deal, Big River, etc). They show up a lot in old-timey honky-tonkin' kind of music. When these secondary dominants are present, you can get away with just kind of playing in C/Am over the whole thing, but you'll find yourself much closer to JG's sound if you approach each of the chords individually. In particular shoot for the thirds of those chords--that is the note that really sets up the following chord.
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Ideal practice tools for improvisation...backing tracks, lessons on modes, CAGED, etc.