In reply to JuneBob:
I have 2 pairs of Mizuno Waveriders which claim to rely on a springy bit of plastic for cushioning rather than foam. They do seem to cope well in the wet (on road) but of course this may be confirmation bias.
I wouldn't take the road shoes on a trail in the wet because the heel clogs with mud & stones (and snow in my experience), but they do a trail-ish shoe (the GTX) which might suit your needs.
Running in a sodden foam-cushioned shoe is likely to seriously degrade its lifespan, though I know in the UK it is hard to avoid.