Mission accomplished!
I got the replacement capacitors in the mail this weekend, spent some time desoldering and cleaning up the connections on the board (I'm SO glad I had the hot air gun for that!), soldered in the new caps, and now it runs like new! I ran HeavyLoad stress test for a while to make sure it's stable, and there were no glitches whatsoever.
Well, now I can do that...
Sky