This is just psych war. They are trying to make you believe that they actually have a legal basis for their claim, which they do not. They cannot retrieve any copyright claim from the university because there is no copyright claim. There was only an intellectual copyright that immediately transferred to you the minute the order was completed through the submission of the document to you. They only submitted a model paper, which was used as the basis of you original presentation. Since you did not use their essay in totality, and you can prove that you had to edit large chunks of the essay, using only their quotes, they cannot claim that they wrote the paper for you either.
It is legal to ask for research help, even pay for the help if needed, provided you have not used the actual paper they submitted to you for a grade. I believe you have nothing to worry about. There are really no worries on your part since they are nothing more than online blackmailers. You can ignore them if you wish or, you may approach your professor and come clean, saying that you used a research service, but wrote the paper yourself. Preemptive movements could help in this case, just to clear your conscience.
These companies always speak as if they are large conglomerates with a battery of lawyers (they do not), they also try to make you believe your have broken laws, which you did not. They only have another employee who knows how to draw up legal sounding and threatening communications. Do not give them any power over you. Ignore them. IMO, that would be the best move for you to take.