So so far the consensus is to unclog the existing drains. I hadn't actually considered this. I have tried to pour water from my (pretty low flow) garden hose and it does overflow during this test. I just climbed up a ladder to my roof. I could take the gutter apart, it connects to the drain at about knee level. I guess I'd assumed that the drain would be un-clogable, but I can check if I can see anything near where my gutters meet this drain.
I should point out that there are no leaves at all in my gutters, I don't have any trees close to the house so I fear whatever's clogging these drains is a more insidious.