Beware is an English verb.
Beware is a lexical verb, but uniquely in English for a lexical verb, it is a defective verb. "Beware" is usually used as an imperative, e.g. "Beware of the dog". It lacks other forms. Other forms often start to feel grammatically weird.
It is a compound verb with "be" as its head (contrast "become" where "come" is the head and that works fine). Given how irregular "be" is, this probably the source of all the grammatical weirdness. Indeed, copular be with the adjective "wary", i.e. "to be wary" seems to be used instead. In that case, "beware" is generally preferred to "be wary" in the imperative, otherwise "be wary" is preferred.
Some older, or poetic texts, may use forms such as "bewaring", "bewared", etc. But for teaching purposes these are probably best ignored.
- at least we can't think of any others
- Julius Caesar Act 1, scene 2, 15–19,