You can do this yourself, if you are determined to do so. You will need to go to court to obtain an order requiring the birth father to submit to a paternity test; a paralegal can help you fill out the proper paperwork for less than an attorney would charge. Once you get the order, you wait on the father to comply. Then you pay for the paternity tests, which will costs about $700 per child (it does in my area, anyway). Then, the father must sign to voluntarily sign away his rights as a parent and the court must allow this. Then, you get an order allowing the adoption and the changing of the birth certificates.
You may want to check on your state's laws as regards a certain period of time without contact from the father being grounds for allowing step-parent adoption. You still have the issue of establishing paternity in order for the father to relinquish his parental rights.
Chances are the father isn't going to get an attorney and spend the money required to fight you. Then again, there are grandparents and aunts and uncles who may not want to lose their rights to be a part of the children's lives and would be willing to give him the funds necessary to fight you.
You could head this off at the pass by entering into a legal agreement to allow the children to have visitation with the extended family members despite the father's allowing the adoption.
If you don't have at least $2,000 available to do this now, then you need to wait until such time as you can afford to take these actions. In the meantime, there is nothing to prevent your children from using your husband's name. Any time a birth name is required, you can simply state 'new name' AKA 'birth name'.
Please never make these children feel that their birth name is an obstacle to your happiness with them as your children. They are who they were born and they are members of this birth family in many ways that have nothing to do with the law's opinion as to whose family they belong. Your acceptance of them as individuals born into family X is of great importance, even more so than being adopted by their stepfather.