OFFHAND, OFF HEAD, BY HEART

English for Today by Ganiu Abisoye Bamgbose (Dr GAB)

Offhand means to be unfriendly, and to show little interest in other people in a way that seems slightly rude:
She isn’t usually offhand; perhaps she wasn’t just in a good mood.
Also, if you say something offhand, you say it without checking the details or facts of it: I couldn’t give the figures offhand (it means I can’t go ahead to give the figure without checking the record).

By heart: If you know something such as a poem by heart, you have learned it so well that you can remember it without having to read it:
I know the whole passage offhand. ✖️
I know the whole passage off head. ✖️
I know the whole passage by heart. ✔️

Off head is not a lexical item in English but If you say that someone is off their head, you think that their ideas or behaviour are very strange, foolish, or dangerous: He’s gone completely off his head.
GAB