Does a teacher have the authority to search me just because they think I might be carrying a weapon or drugs?.
It depends on whether the teacher has a really good reason to believe that you are in possession of a weapon or drugs. A school official can search a student if he or she reasonably believes the student is in violation of a school policy. First, a school official has to have a strong suspicion that you are in possession of something you are not supposed to have at school. This means that the official has to have more than just a hunch. For example, you cannot be searched just because of the kind of clothes you are wearing. However, a teacher might be able to search you if he or she hears other kids talking about a gun they saw you with, or if the teacher smells marijuana when you walk by.
Second, the teacher can only search you if the search is necessary to maintain discipline or enforce school policy. For example, a teacher probably cannot search you for carrying cigarettes, even if you are not supposed to smoke them at school, as long as your possession of cigarettes is not causing any substantial disruption and you have not been or are not getting ready to smoke them against school policy. However, the teacher might be able to search you for cigarettes if he is seen you smoking against school policy or he has good reason to believe that you are on your way to smoke on school grounds.
Finally, a search has to be reasonable in terms of a student's age, gender, and the nature of the suspected violation of school policy. For example, strip searching a student for a suspected theft of candy from the cafeteria is probably unreasonable. However, a teacher probably could ask you to empty your purse or pockets in an attempt to uncover evidence of theft. If a student is suspected of a more serious violation, like carrying a handgun, then a strip search might be reasonable. The teacher should not search you in front of other students, and the search should be conducted by a school official of the same gender as the student.