Mushy courgettes

I've recently moved to East Scotland and am trying out my new garden and new climate - dry and windy. I've planted various varieties of courgette but am having problems as once the flowers are dying off, they then rot and the courgette rots too.  Any ideas, please? I'm removing the flowers as soon as they've bloomed but that seems such a shame and it seems to be prevalent on round courgettes rather than the long ones.  Thank you

Yes, both male and female.  The flower dies and rots and then the insode of the courgettes or the whole thing just goes to mush - yuk!

They seem to have pollinated ok and some drop off of their own accord - which is rather handy as they then don't rot! 

Ah, now I understand.  If it's been pollinated then the flower drops off and the courgette grows ok.  The ones that appear to rot with the flower attached havn't been pollinated.  It happens to mine too but I get round it by pouncing when a male and female flower at the same time.  I like to think of it as a dating agency/fertility clinic.