I don't know if this has been talked about before, but I've never been told about it anywhere.
You're new to handstands. You can balance with the wall, but not for very long freestanding. Well, you're stuck between two extremes. The obvious answer is strengthen your wrists, but there is another wall handstand you can try.
Do a cartwheel into a wall. With your legs straddled, one foot touches the wall and the other doesn't. You'll be working a lot harder to stay balanced because that one foot of contact is on the side of you, not the front. That foot can help with your balance, but because of mechanical disadvantage, it won't be able to help you a lot. Your foot should push forwards and backwards against the wall (sideways technically).
Give it a try!
Oh, one last thing; the straddle of your legs will make the handstand easier. It lowers your center of gravity and makes you more stable. When both legs are straight, you're a little more unstable. Thing is with this method though, you'd be leaning to the side if you had your legs together. That don't fly.