If the domain name is the same as their LLC but I add "by (my name)" to the title of the company, is that enough of a difference?
Enough for what?
I will register it in Ohio with the "by (my name)" added on
Register it with what agency? For what purpose?
Will that suffice, or do I need to start over and operate under a new domain and email address?
It's not entirely clear what you're asking, but your question touches on a few areas of law.
For purposes of discussion, I'm going to assume the name of the other company is Really Cool Widgets, LLC ("RCWLLC") and that the business you started is a sole proprietorship called Really Cool Widgets.
Your operation of a business called "Really Cool Widgets" would likely infringe on RCWLLC's trademark rights, and RCWLLC could probably obtain an injunction preventing you from doing business under that name, including maintaining a website at
www.reallycoolwidgets.com.
Changing your business name to "Really Cool Widgets by Lucy Beth" probably would not be enough to prevent the above.
HOWEVER, since RCWLLC, despite being
listed as an active LLC by the Ohio Secretary of State, apparently is not operating, then it is unlikely that RCWLLC will pursue any claim against you. Moreover, even if RCWLLC wanted to pursue a claim, if it is not doing business, then its trademark rights may be weak or non-existent. And I disagree with the prior response that said you're "almost guaranteed" to be sued (I do, however, agree, that an agreement to purchase rights in the name from RCWLLC wouldn't be a terrible idea). And, while I generally agree with the point that "Zigner" made, I don't think it's a good analogy because of obviously different facts.
Of course, changing your business's name entirely might be the best option in the long run, but I obviously don't have any facts about goodwill you've generated under that name or other things that may counsel against doing that.
If that doesn't answer your question, please feel free to clarify your question and the relevant facts.