If my employer did not include employer sponsored health coverage (Code DD) in box 12 of my corrected W-2, do I need a new corrected W-2

Yes, you should ask for another corrected W-2.

Why? Because at least one state (New Jersey comes to mind) treats health insurance premiums (this is what code DD is) differently than on the federal, so TurboTax needs to see this to make the correct entries on the New Jersey return.

In most situations, it won't change your federal return, but since I don't know what state you are referring to, better safe than sorry.

I am in Tennessee.  We do not have a state income tax.

It would be good to get a corrected W-2, just for your records.  Did you have access to "employer sponsored health coverage"?

Since you're in TN, a corrected W-2 is not necessary for tax purposes.

Yes. On the original W2, the amount for Box 12a Code DD was listed, but my 403b contributions were not.  On my corrected W2, the 12a box included the 403b contributions, but the employee sponsored health coverage was no longer included in box 12.

Do I need to get another corrected W-2 before I file my taxes?

No, you do not have to get another corrected W-2.  The code DD in Box 12  is for informational purposes only.  The amount shown is not included when you prepare your tax return.