Excellent question, and you are right on. せんせい is honorific, and is not the best word to use when referring to yourself. However, in the context of learning Japanese as a foreign language, and in situations where vocabulary is limited, it is acceptable to use せんせい to convey your meaning. As your Japanese level increases, of course, then your modes of expression will as well - this kind of situation is not limited to the word せんせい, either. But to answer your specific question, yes. きょうし (教師）is another word for teacher that does not convey any honorific meaning, so you can say: わたしはきょうしです。
Sensei is a kind of title and also can be used to address lawyers, politicians, masters of martial arts, etc. Kyoushi is an occupation, generally schoolteacher from elementary school to high school.
I think that わたしはきょうしです (I am a kyoushi) is better for communication among adults. However, it means that "My job is teacher", so we don't say, "わたしはこのクラスのきょうしです" (My job is kyoushi of this class). Instead, we say, "わたしはこのクラスのせんせいです" (I am a sensei of this class). Of course, you can say, "わたしはこのがっこうのきょうしです" (My job is kyoushi of this school).
In informal casual situations, "～がっこうのせんせいです" (My job is sensei of ～ school) would be better.
In addition, kyoushi usually refers to a teacher of elementary and secondary education with government-issued certificate.