You could but it's not really elegant, or very usual. It's certainly not a case of a form of words being acceptable in formal English, which would sound starchy elsewhere. It would sound a little odd anywhere.
I puzzled over these sentences for a while before I realised why I think they are odd. It's the structure. They are both: I was <adjective thing> because... .
It's most disturbing in the second sentence: I was <adjective thing> because I was <adjective thing>.
It would be more natural, to me anyway, as: I was rushing because of my lateness. I was rushing because I was late.
And I prefer the second.