The 1098-T should cover everything that the school billed you or your daughter whether it was paid by grants, scholarships, cash or loans. So, all you need to do is enter the 1098-T as it appears, there is no place special that you need to indicate out of pocket payments to the school.
Yes, that would be double dipping so no, you should not enter it a second time if you are already getting a credit in MA for the amount that is included in your federal return. To be sure that you are getting the tuition deduction for MA you can look at your MA Schedule Y like 11. If you do not see the credit, then you can add it to the state section, however, you should already be getting the credit based on the federal carryover information.
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