If stopped working midyear, should I still check 'still work at <previous employer>' when filing tax?

Those questions you are looking at "still working at X" are only used by Turbo tax to pull some of the basic W-2 info from the previous file.

Because you had income in 2025 from the school you will get a W-2 from the school. later Turbo Tax will ask you if you or your spouse had any other w-2 forms. That would cover the case where you did find a job in 2025.

Next year (unless you got rehired by the school) you will answer this question No.

The question and answer is designed to make it easier to complete the forms.

They will ask you about your bank so they can pull the basic info for your 1099 forms.

Is there a proper way for me to become a dependent of my spouse for the tax filing purpose?

You aren't a dependent. You are married and will either submit this year (or any other year) and married filling separate or married filing jointly.

Later in the process Turbo tax will help y0u decide which one makes the most sense.