use of "bad mouth"

Please let me know if it’s grammatically correct:

  1. Just because I committed one mistake that doesn’t mean you should bad mouth about me to others.

…doesn’t mean you should bad mouth me to others. (informal)

I thought ‘badmouth’ all in word.

As it’s informal slang I don’t believe I’ve ever seen it written down before, but I would never consider it to be one word. Google has both - perhaps we can compromise by hyphenating it, as this seems to be the most common form:

Thank you for your explanation.

Had a question why aren’t we using “bad mouth about me” and why just “bad mouth me”? What is the reason of not using “about”? Please explain. thanx

It’s idiomatic, Hardwork. The ‘about’ isn’t specifically needed because it is included within the meaning of the idiom:
‘bad mouth’ = ‘say unpleasant things about’
‘bad mouth about’ would be equivalent to saying ‘say unpleasant things about about’.

So is this correct

  1. She was bad mouthing you

Yes. “She was saying unpleasant things about you.”