I didn't use nursing pads at night, I had a stack of old t-shirts I used specifically for nighttime nursing babies. At 12 weeks, you and the baby have found a rhythm to your nursing relationship, but the amount of milk you produce will change with her needs. Early on, these changes take a while to take affect on your milk production, and your baby may only be sleeping like this temporarily (although for your sake, I hope it's permanent! I am so jealous!) which is why you still have so much milk at night.
Hang in there, your boobs will get the hint pretty soon. Just make sure you aren't wearing anything too constricting at night, because you then run the risk of a clogged duct and that could lead to mastitis. It would probably help to express some in the morning if she doesn't want to eat right away or if you wake up earlier than her. Not too much, or you will signal your body to produce more, express just enough to relieve that "so full I'm gonna pop" feeling.