If my adoption is not finalized yet, can the 5 year mark start next year once it is finalized? Or do I have to claim it this year?

claim credit for expenses paid 

 

domestic adoption

before finalization - in the following year

year adoption is final and any paid in succeeding years  - year paid

 

foreign adoption

before finalization - year adoption is final

year final or following years  - year paid

 

it is the credit that has a 5 year carryover.  thus any unused credit from 2020 could be carried forward until 2025