Problem solved!
I finally had a moment to replace my 42-strand wires with a new set of 16-strand Gibraltars. While I was at it, I swapped out the old Ambassador batter for a new Fiberskyn, and man! What a difference. The drum used to be unplayable, but now it's sensitive, warm, rich and woody. It's still an acquired taste, and very different from the snare drums I'm used to playing, but it's a beautiful sound. It reminds me of Art Blakey's snare on "Caravan." Now if only my playing sounded like Blakey's, I'd be in business. :) Thanks for the tips, everyone.
If nothing else, this experience has convinced me that 3-ply Gretsch snares were meant to be played with calf heads. The Fiberskyn is a nice substitute, but if I ever get around to it (i.e. when my daughter is grown), I might try tucking calf heads for this kit.
Despite the success, I'm still finding that I have to crank the snares tightly enough that I hear the snares even with the throw-off disengaged. Does anyone know a solution to this problem?