The full $1100 refund from state A will be included as taxable income. The balance due paid to state B can be entered and included as part of your itemized deductions for 2023 (subject to the $10,000 limit for state and local taxes). So, in the end, they may offset one another, but it is not reported that way on your tax return.
Go to Deductions and Credits > Estimates and Other Taxes Paid > Other Income Taxes > Other Income Taxes Paid in 2023 > Balance due on your 2022 state taxes paid in 2023 and click Start or Update to enter the balance due amount you paid for state B.
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