why are some of my courgettes rotting on the plant?

I'd replace the soggy straw with dry straw in any case as this won't do the plant any good and could lead to mildews and rots. If the soil is already wet then a dry mulch will keep the humidity in the ground. If the courgettes are rotting while still very tiny then this is often just a case of the fruit not being pollinated. Often too, the female flowers start rotting and this goes back into the flower end of the courgette. I'd pull of the flowers once the courgette is clearly growing away. Once they get going you should have a courgette every three or four days.

I get this every year, but it always eventually sorts itself out. As you say, the wet weather isn't helping things.