If box 1 of both of your W2s added together is $82k, and you file single, then you should have a total federal tax liability of $11,339.50.
$82,000 Form W2, box 1
-$12000 standard deduction
= $70,000 taxable income
$9,525 taxed at 10% = $952.5
+$9,526 to $38,700 taxed at 12% = $3501
+$38,701 to $82,500 ($70,000 in your case) taxed at 22% = $6886
= $11,339.50 total
How much was withheld (box 2 of your W2)?
State tax is another matter, and can vary depending on how each state handles part year residents and how long you were a resident of each.
The federal withholding tables did estimate low for much of the year, I think, so it's not too surprising that you owe. I'm actually a little surprised you owe as little as you do. That's pretty good.
You might consider switching those Roth contributions to traditional unless you expect your expenses in retirement to go up over $95k if you stay single or 190K if you get married.