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For high school, if you use the recommended procedure in the rule book for overtime, then the ball is dead immediately if team B gains possession. It is however possible for team B to score a safety or a touchdown. Team A fumbles in their own end zone and B recovers in the end zone, for example, unlikely but possible.
Each state association can adopt its own overtime procedures, so your state may vary. Are there any states that use NFHS rules that use a different overtime procedure???
In the NCAA, the ball remains alive after a change of possession until it is declared dead. So it is possible for B to score and get the results you described.
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