Yes, you qualify as a full-time student as long as you attended school for at least five months (they do not need to be consecutive) and were enrolled for the number of credit hours that your school or program considers to be full-time.
It is possible to be both a full-time employee and a full-time student in the same year.
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And what are you trying to do with that information? If you are over the age of 23 then being a full-time student will not help your parents or anyone else claim you as a dependent since as a full-time teacher you must be earning over $4200 per year.
As for education credits you can explore the Lifetime Learning Credit.
Go to Federal>Deductions and Credits>Employment Expenses>Teacher (Educator) Expenses
Each school has their own definition of a full time student, and the IRS goes by what the school says. So only the school where you are enrolled can clarify that status for you.
If you'll look on your 1098-T (if you received one) there's a checkbox in box 8 for "check if at least half-time student" Some colleges will check that box because if you're a full time student, then you are "at least" a half time student. Others will not check that box. But not checking it doesn't mean you're not full time.
Generally if you are working a full time job, it's extremely difficult, though not impossible to be a full time student too. So that's why you need to get clarification with your school.