I am not sure I understand. Your DH(BTW, what is a DH?, sorry to ask).
Your DH is going by an alias and you used only the newspaper to announce but, it is not a legal name. It is an easier name to pronounce and posting to the newspaper cost only a couple of bucks?
The route of changing your name and identity by legal means is expensive and time consuming you are required to post your new status in a newspaper anyway. This is not a good analogy but think of even a sex change operation. Your current is Michael but due to your sex change you need to create a new identity and you want to be Marcia. You can do that and your documents and entities with an interest in your identity will be consistent under the law.
If your DH did not “legally” change their identity, then they are AKA.
AKA costs less and nothing wrong with it, at all.
We did look into it carefully - there’s a line of books of legal self-help books called Nolo Press that we used, specifically for the state we lived in. According to the book we followed, there were two legal methods - doing through the court, or doing with a specific form - included in the book - getting it notarized, and then sending an announcement (form also in the book) to the newspaper a certain number of times. Also, DH (that’s Dear Husband!) had to obtain a new passport with the new name within a certain time frame, and had to immediately start using his new name on every paper/account/bill etc that had his name on it.
(DW is Dear Wife, DD is Dear Daughter, DS is Dear Son, etc)