I'm no good what-so-ever at critiquing, but your elbows need to be bent more, having them like this throws your seat forward. I'm quite guilty of this also and need to remind myself constantly. Great for first time in a year though!
Your hands are very open - your fingers are lightly grasping the reins with the very tips, not actually wrapped around them. Hold the reins - in a fisted hand, not with fingertips. This gives you better control of your hands. Additionally, you need to keep your thumbs on top.
I think dropping your stirrups will help with your posting and balance. Remember, don't grip too much with the back of the calf, but the side. Be careful not to grip with your knee! Also, when she begins to trot do not lean forward. This puts more weight on the forehand, throwing you and her off balance and encouraging her to walk to regain her balance. If I was riding a horse that was only giving me that much trot, I would have given her a hard kick and if that didn't work I would start carrying a crop!