This doesn't make sense.  Taxes cannot be withheld from a 403(b) loan because the loan is not a distribution.

This where I hope it was TIAA's mistake. $3,000 withheld for federal on $15,000 loan.

If tax was withheld you must get a tax document to show that.

If it is a loan assemble your proof documents .

You can request a corrected 1099-R if there is a discrepancy.

how much cash did you get? 15 k or 12 k ??

This is something that you'll have to sort out with TIAA.

It's quite unusual that a plan would permit a loan to be initiated from the account of a former employee.  Most plans only permit current employees to take out loans.

The fact that $3,000 was withheld for taxes seems to imply that this was a $15,000 distribution with the mandatory minimum of 20% withheld for taxes and not a loan, with the deadline for rolling over the distribution normally being the 60th day following the date of the distribution but because of the COVID disaster declaration that deadline would have been extended to July 15. 2020.  But that wouldn't explain why you would think that you had a 5 year loan and were able to make monthly repayments.  If this was indeed a distribution, the auto-payments made by July 15 could be considered to be rollover contributions, but any beyond that would be excess contributions subject to penalty.  Again, you'll need to sort this out with TIAA.

Yes, a little less than $12K. About $300 withheld for state, too.