Turbo Tax show I still owe additional amount after I already paid the FED AMT due by the deadline I don't understand why I still owe when I got refund on the state. HELP?

Where are you seeing that TurboTax shows you still owe tax?    Your TurboTax account will always show that you had tax due if you prepared your return using the software.    TT never knows how, when or even if you paid your tax due.   Your account will only show the result of the tax return you prepared.  Your proof that you paid will be your own bank or credit card records.

Did you pay the federal tax when you filed the tax return, or before you filed, or after you filed? How did you pay it?

Does the federal tax return that you filed show an "amount you owe" on Form 1040 line 37? Is that the amount that you paid?

And....the fact that you received a state refund has no bearing on whether you owed federal tax.   Your federal tax due and state refund are completely separate---one has nothing to do with the other.