Yes, if your 19 year old's primary home when he is not at school is your home, then you would be the one eligible to claim him as a dependent on your return and claim head of household. In order to claim a child, the custodial parent is the parent who the child lives with more than 6 months. Being away at school counts as temporary absence so it still counts as being at home and if that is your address, then that would go to you.
For the 1098-T, you may be able to contact the school to get a new one or it may be available on their website in your sons financial aid section.
If his mother does claim him and you do as well, then the IRS will use the Tie Breaker Rules to determine who gets to actually claim him. Depending on who files first, if he is claimed on both return, the second to file would need to print and mail there return as it will be rejected for duplicate use of a Dependent, unless you use an IP PIN. Then you would still be able to e-file. Since he is 19, he no longer qualifies for the Child Tax Credit, however, you can still get the $500 Non-refundable Other dependent credit.
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