In 2019 my husband legally adopoted his stepson. We started the paperwork in the fall of 2018 and was finalized in 2019. Do I need to claim the adoption expenses?

Please clarify.   Do you mean that your husband adopted YOUR child?   If so, you cannot deduct adoption expenses.  

No, according to the IRS,  An eligible child is an individual who is under the age of 18 or is physically or mentally incapable of self-care. Qualified adoption expenses don't include expenses that a taxpayer pays to adopt the child of the taxpayer's spouse.  

For more information, See IRS Topic No. 607 Adoption Credit and Adoption Assistance Programs