I have boney, skeleton fingers, with big knuckles, and I find I need to position the bumps in the right place. Now, for F you dont need the G string down, because it is fretted higher up, but when it comes time for F minor, the G string will be open, so it is necessary. I find that if I roll my finger on its side and slide it upwards, like pointing it up towards the sky, the G string gets coverage.
It is basically, these are your hands, and you know what you have to do, which is cover all those strings well. Now, you have to find the way to make the hands you were born with do it. It is as easy and as hard as that. Your fingers need to press down the strings nicely. Make them, and then practice doing that quickly. I dont know any other way.