Well first and foremost: congrats on sticking it out and trying to find out how to make it better instead of giving up! That's half the battle right there. As for making it more stable: not sure what you're asking. When I stand up in the pedals I'm not only leaned way over the handlebars but I rock the bike from side to side especially if I'm climbing. Is that what you were talking about? Also, being a newbie rider and climbing a monster hill not once, not twice but 4 xs makes you a super-woman in my books!
PS: welcome to the board and glad you're finding it as fun as I do! If you get a chance, there's a thread under the open titled "getting to know you". Gives you a chance to learn more about us and tell us more about you.
Life should NOT be a journey to the grave with the intention of arriving safely in an attractive and well preserved body, but rather to skid in sideways, champagne in one hand, strawberries in the other, body thoroughly used up, totally worn out and screaming: "Yeah Baby! What a Ride!"