Another head-scratching update: I could not reproduce the distortion using my phone to feed a low level music signal through the Eros. The EQ was horrible, of course, but the sound was clear.
One thing I noticed is that one of my Sowter 1990s has half as much secondary resistance as the other (860Ω vs 1.4KΩ). I find that alarming but I didn't notice the channels sounding different through headphones when I listened to just the left or the right channel. So I don't think the distortion is coming from one of the Sowters. (But I wonder if I should contact Sowter and ask about the impedance mismatch.)
In fact, I don't think the distortion is coming from the Eros. When I play records, the distortion is now constant, but I didn't hear the distortion in the music played through the Eros from my phone.
I found an old Koetsu cartridge in my box of used cartridges that still might have some life left, so I installed it on the Linn LP12 and the distortion was quite noticeable, so that rules out the new Hana ML I bought. I replaced the tonearm cable from my Linn tonearm, just to test that possibility (I have two stock Linn tonearm cables, and they can be fiddly and need to be installed just so). No difference.
Could the distortion be coming from my LP12's tonearm? It's a Linn Ekos and the only thing about it I don't like is that it has a very small headshell area and that makes fitting cartridges difficult. The Hana wanted to be mounted back further in the headshell than other carts, and I ended up very slightly bending (or so I thought) two of the four cartridge posts on the Ekos--you know, the gold plated pins that emerge from the tonearm where you install cartridge leads. If I disturbed the wiring connection inside the Ekos, could that explain the distortion I'm hearing?