No. You are not be paying taxes on your 2021 unemployment income on this year's return.
If you have W2 income, you may not have enough taxes withheld. Also, the tax brackets were increased by 7% from the 2022 tax year because of inflation thus the IRS made inflation adjustments to the 2023 tax rate schedules.
You may wish to increase your withholding amount from your pay so you won't owe taxes next year. You can do this by completing a new W4 and submit to your employer. Here is a form you may download and complete. In this form, there is a worksheet in the form you can use to determine the correct amount of tax withholding that needs to be made.
Inflation Adjustments