Instead -- and maybe because I planted too thickly -- the corn
is going gang busters, but is shading the cucumbers. Little cukes are
forming, but I am concerned about the perceived lack of sunshine.

Yes, a common issue with growing vines up corn stalks.

In the case of beans, there are bean varieties which are specifically
advertised for being more shade-tolerant than usual, just for growing
up corn stalks (for example, at http://www.southernexposure.com/ ).

Not that this helps you before next year, of course...