Leaving my husband....

I have been with my husband for 4 years. We have a 2 year old together. When we first got together my older daughter was 3. She is now 7. She has begged my husband to adopt her and her biological dad has agreed to sign over his rights because he is in jail for a very long time. My husband said yes. My older daughter wants the same last name as myself, my husband and her little sister. So he starts the process and it gets "too hard". He is being lazy. He had a whole week off of work and not once did he do anything regarding the case. He is dragging his feet. I have given him many chances to back out and he just has a bunch of excuses. So today I told him if he doesn't get this shit done by the end of the month I will file for divorce and we can figure out custody of our little one. Am I overreacting? I don't feel I am. I won't let him lie to my older daughter and be lazy. If he won't include her in our family, I don't want to be married to him. If she isn't worth fighting for, I don't want to be with him. My children come first. He is a good dad (minus always playing on the cell phone -not talking to other girls, playing stupid games). I'm lost and confused. I don't want to break up my family, but I can't let him do this to my daughter....