penalty after amending

Under IRC code section 25A(g)(3) no credit is allowed to an individual who can be claimed as a dependent on another's return.  your dependent children are not entitled to claim the credit.

your kids' 1098-T goes on your tax return and you collect the tuition and expenses credits.

If you didn't do it that way you only have to amend one tax return, yours.

When I check the boxes for can be claimed as a dependent/not being claimed as a dependent, TT adds the credit.

Yes.  Must be claimed as a dependent to claim the credit.

Qualifications for claiming the American Opportunity Tax Credit are:

• You paid an eligible student's qualified education expenses for higher education at any college, university, or vocational school with a student aid program administered by the US Department of Education.
• The eligible student is you, your spouse, or a dependent on your return.
• For the full credit, your MAGI (modified adjusted gross income) is less than $90,000 ($180,000 if you're filing jointly).
  • For a reduced credit, your MAGI is between $80,000 and $90,000 ($160,000 and $180,000 if you're filing jointly).

An eligible student is defined as a student who:

• Is enrolled at least half-time in a program leading to a degree, certificate, or other recognized credential.
• Had at least one academic period beginning during the year.
• Didn't claim the American Opportunity Tax Credit for more than three previous years.
• Didn't complete the first four years of post-secondary education before the beginning of the year.