I would like to tell you, that after getting divorce from your ex-husband, now your are no where related with your Exhusband. So by marrying with another or be making yourself as a life partner of another by valid marriage, you are legally wedded wife of him.
So far as the question about the adoption of child which you begotten from your Ex-husband, it does't matter for the Adoption Point of view.
As per your Divorec decree, if u have custody of your / subjected child, then by way of Decree, u are Natural Guardian of that child.
The main point is that, Adoption proceure should be by way of Adoption Deed, and after Adoption there should be Publication in the Official Gazzette about the change in the Name of that child.
More over for the safer side, u can publish the same in the News Papers circulating your local area / city as well as at the place where (as per your Knowledge) your exhusbad is residing.