Thanks for the great suggestions, all. The Continental 4000 tires I run are kevlar, and I go for several months with zero flats (replacing them about 2000-2500mi), but soon as they sand the roads...it's another story. That's why I wondered if it's something to do with the sand. Damn sharp sand, it seems.
I will check the rim tape, though. (That's not something that I was aware of needing to check. ) What does worn rim tape look like, BTW??
I considered those Slime tubes, Tvash, but they didn't come with the right size valve stem for my tires. But I ordered some Slime Tire Skins and Forte tubes that said "Thorn-resistant."