Regarding the adoption credit, the IRS has stated that an eligible child is an individual who is under the age of 18 or is physically or mentally incapable of self-care. Given the age limitation, that probably explains why you are not seeing the credit for your children who are 18 or older.
Tax benefits for adoption include both a tax credit for qualified adoption expenses paid to adopt an eligible child and an exclusion from income for employer-provided adoption assistance. The credit is nonrefundable, which means it's limited to your tax liability for the year. You are correct in that any credit in excess of your tax liability may be carried forward for up to five years. The maximum amount (dollar limit) for 2021 is $14,440 per child.
Here is a link to the IRS webpage that discusses the adoption credit which you might find helpful.
Adoption Credit
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