I received a 1098-T form for 2016 for tuition that I paid off in 2019. Can I claim that expense for 2019 since I made the payments in 2019 not 2016?

Unfortunately, no.

From the IRS website regarding timing of the expense.

Academic Period

You must pay the qualified education expenses for an academic period that starts during the tax year or the first three months of the next tax year. Academic periods can be semesters, trimesters, quarters or any other period of study such as a summer school session. Academic periods are determined by the school. For schools that use clock or credit hours and do not have academic terms, the payment period may be treated as an academic period.

You may, however, be able to claim the interest paid.

Entering a 1098-E in TurboTax:

Here's how to enter your student loan interest. Follow these instructions whether or not you received a 1098-E from your lender:

• Open (continue) your return in TurboTax if you don't already have it open.
• Search for 1098-E (upper- or lower-case, with or without the dash).
• Select the Jump to link in the search results.
• Answer Yes to the question Did you pay any student loans?
• If you land on the Here's your 1098-E info (or Here's what you paid in student loan interest)screen, you can either edit an existing payment or add a new one.

Follow the onscreen instructions.