open-note essay exam with the questions given ahead of time, what is considered cheating?
I think the other answer is a bit more scolding than is necessary.

Your professor has your email to your mom, didn't comment on it, and forwarded it to the person responsible for recording grades. It sounds like what you are worried about is not a worry to the professor, and it sounds like your professor is quite a reasonable person where technical issues are concerned (i.e., by accepting your evidence that the work was done).

If I were in your position I'd concern myself more with making sure your submissions are fully accepted in whatever software you are doing the submission with in the future - hopefully the software provides some sort of feedback or way to check.

You can also consult the course syllabus to see if there are any specific policies on what is allowed and not allowed for exams. I don't think you are crossing any lines by asking for proofreading help (presuming you are not taking a course where proofreading is a major skill that is being taught). I also don't think it is necessary to explain yourself to your professor unless you find a written policy that makes what you did clearly against the rules: by sending the email, you haven't made any attempt to hide what you did so there isn't really anything to come clean on.

Besides that, I would stop worrying about this and focus on any other studies and responsibilities you have. You're doing fine.