Courgette seed saving

I think you'll need to leave the courgette to fully mature which will stop it making more courgettes.
I would personally pick this one and eat it and use it as practice for hand pollinating. Then towards the middle of the end of the season I'd hand pollinate another one and let the plant fully grow that one on to maturity. Then harvest and store it for a while (you'll effectively have a marrow), after a week or so open up and scoop out seeds and follow the advice above for pumpkins.