If you are going to owe, then yes you need to file an amended return. You will do this because (1) your return is wrong (even if it was not your fault), and (2) interest on the unpaid tax will continue to accrue until you pay the tax.
You should file the state amendment if it has been several weeks since you made the original filing. Normally, they don't want you to amend until the original return has been processed, but this year, it may be very difficult to know when they might be.
So, just file the amendment and cut off the interest on the unpaid amount.
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