Advice: you probably will lose your claim because you are barred by the doctrine of contributory negligence!
I hate giving bad news, but it appears very clear that you will lose your claim. The reason is that you were at least partially at fault. In a contributory negligence state, a victim who is even 1% at fault is not entitled to recover any compensation at all.
At this point, you have no choice but to (a) obtain an agreement from the other dog owner that he or she should pay you, even though there is no responsibility to do so, (b) proceed against your pet insurance, if you have any, or (c) accept what has happened and stop attempting to collect from the other dog owner.
There are no further steps for you to take, other than those mentioned above. There is nothing in Volume 1: Text or Volume 3: Forms that can help you in your state.