Also took it by the local tire & suspension place for an inspection, and they said it looks solid - even has modernized (and new) bushings and other suspension parts that are grease-able. Good news!
About the way the mufflers are hung (straight-through glasspacks that dump in front of the rear axle), he said, "This is cute, but I've seen worse." They just kind of dangle from straps. They won't fall off or anything, but clearly making it high-tech was not on the prior owner's mind ;-)
He suspects the loose handling is due to the rear end being jacked up and having big tires out back and small ones out front. He said the springs could be replaced, but they don't look shot, and that just putting on reasonable tires will go a long way toward making it drive better. Needs front wheel bearings, but that's about it.
Woohoo!
And now: New photos!

I am in love with this angle. I mean, LOOK AT THE SHAPE OF THIS THING! A timeless and beautiful design.

This car has no bad angles. Look at that great ass!

In front of the house (the blue one on the right).

Love interior design from the 60s!

ROAR! Here dwells the 396 big-block Chevy engine, now a 408 with beefy cam driven by gears (instead of chain), ported & polished heads, high-performance exhaust headers, and lots of other goodies. About 450 rear-wheel horsepower.

Front end closeup. LOVE that nose!
I'm in love.
Chris