You need to pay state taxes separately. The IRS and the state do not share the money. They are separate entities, so if you do not pay separately, the one that does not get the money will penalize you.
You only need to pay the quarterly taxes on the amount you earn for your gig work. However, to determine your tax rate, you will need to consider her income and your other income as well.
Depending on how you want to do it and how much you make with your gig work, you can also change your W-4 to increase the amount you have withheld to cover the extra income. This would allow you to not have to pay the quarterly payments as long as you do make sure you are having enough extra withheld.
What is the self-employment tax?
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