Enter amounts shown in box 12 of your W-2 only in box 12 of TurboTax's W-2 form, nowhere else in TurboTax. The items in box 12 pertaining to your 403(b) contributions will have code E or code BB. the amount reported with code E has already been excluded by your employer from the amount in box 1 of your W-2 so you don't get to deduct that again.
I don't see how that's possible. Are you sure that the increase in balance due was not simply due to the amount in box 4 of the W-2 not being sufficient to cover the tax liability being added by the amount in box 1?
Try going back and seeing if changing the code E amount to $0 decreases your balance due. If it does, compare your Form 1040 with and without the actual code E amount to see what is being affected.