I just moved to Ohio in June 2023 but Ohio is asking me to account for the full year residency but there's no where to enter the info on the form. What should I do?

After indicating you Ohio residency as part year, TurboTax (TT) will ask for the dates. But, that doesn't automatically carry to the SD return   In addition to state tax, Ohio has both city and School District (SD) Income Tax. Not everybody is subject to these local taxes. SD 100 is for the  School District  Income Tax (SDIT). SDIT does want your dates of residency (new this year) First be sure you're subject to the tax. 

 To verify or find out if you live in a school district or city with an income tax enter your address at:

http://www.tax.ohio.gov/Individual/LocalTaxInformation.aspx

TurboTax (TT) does not automatically prepare a School District return (form SD 100), you must initiate it.

At the screen, in the Ohio program, titled  "A few things before we wrap up your state taxes"
      scroll down to School District taxes. 
Or later at the screen  "Your 2023 Ohio Taxes are ready for us to check"
    -Click Other situations
    -Scroll down to School District taxes.

You may have to go back through the SDIT interview to find and edit the dates screen.

Be forewarned, the dates entry is convoluted.  For your school district, you will enter 06/xx/2023 to 12/31/2023. For non residence you will enter SD#9999 and dates 01/01/2023 to 06/yy/2023. 

xx is the date in June you first resided in that SD and yy is the day in June before that date.