Health insurance change

So I am looking for some opinions on my situation. I am due oct 10, my new insurance plan kicks in oct 1, but i am worried I may delivery before the new one starts. My partner and I were planning to marry at some point, but had no immediate plans made when we found out we were pregnant. And I have shitty insurance at the moment. we are considering going down to court house to married so that i could change my insurance before oct 1, in case she decides to come early. Part of me is sad to rush the whole marriage thing, but if I deliver before oct 1 its going to be super expensive, like literally ten time what it will be under my new plan. What should I do??