I lived abroad and paid foreign income tax in spring of 2023, I moved back to the states and worked in Texas. Do I need claim my foreign income tax, and if so how?

You may qualify for a foreign tax credit for the foreign taxes you paid. To see, choose the Foreign Taxes option under Estimates and Other Taxes Paid, in the Deductions and Credits section of TurboTax:

Indicate that you need to complete Form 1116 when asked. It's usually better to take the foreign "Credit" option over the "Deduction." Choose the countries you paid the taxes to and enter the income you earned there and the taxes paid. You will get a credit for foreign taxes to the extent you had US tax on your foreign income. The balance can be carried back one year and forward up to ten years.

You need to report your foreign income as well. You will do this the same way you report income earned in the US based on the type of income you earned. If you earned wage income but did not receive a W-2 form, you can report that in the Wages and Income section of TurboTax, then Less Common Income, then Foreign Earned Income and Exclusion. You don't have to apply for the exclusion of income to report your foreign income in that section.