So at the last day of class a classmate asked if they could borrow some of my lecture notes for studying for the final (this was a history class). I said sure and sent them to their usb. Since it was near the end of class I was rushing to copy and paste my files and in that I think I also sent them my essay notes by mistake. I do not know their name and I have contacted the department head and my professor about the issue letting them know as I am worried they could use these notes in their paper. The notes are not my draft or final copy, they just summarise each article the professor gave for the essay, and the paper itself was well overdue at this point. The office told me I would not likely get in trouble but I still have worries as marking is ongoing. Have I done the right thing so far? Is there anything else I should do?
You have taken appropriate action already. If you meet up with the person warn them against using the notes. But, if you want some additional security, print out your notes and get someone you trust to sign and date them. Now you have a clear marker. I doubt that this is necessary, but if you are paranoid,...
But any infraction of rules would be by the other student, not yourself. Notes are just notes. Relax.
Have I done the right thing so far?
Yes. You reported it.
Is there anything else I should do?
I remember once an issue where I moved into some campus housing. There was a container left in my unit's storage shed that was attached to the back of the partment. It turns out that the container was used motor oil. As I toured the apartment, I let my tour guide know that I wasn't worried about it. I didn't know what was in the container, and planned to throw it away. But what really happened is that I never used that storage unit, and forgot all about it...
... until the weekend when I visited my family, and somehow (wind?) the container spilled. Children from nearby apartments were playing in the liquid, so there was a massive clean-up operation. I first learned about it when I unusually looked out the window of my sliding glass door and found a construction barrier placed there, and tape labelled "Do not cross", etc.
I was then getting billed for the massive clean-up effort. To that, I objected.
The school ended up taking my report and researching the issue by contacting the former apartment supervisor who was my tour guide, and learning that there was agreement that this oil originated from a previous tenant. Although it took a couple of agonizing weeks to get an answer, they decided to not charge me.
The office told me I would not likely get in trouble
This is probably the best you're going to get. I mean, it'd be even better to have that in writing. But getting that in writing seems rather unlikely to me. So, you'll just need to hope for the best, knowing that usually such statements made by staff are accurate, and even if they aren't, usually (but not always) an organization will decide to honor such statements if they are validated by the person who said that.
but I still have worries as marking is ongoing.
You can likely feel even more comfortable once your final grades are provided to you. At that point, a school can likely still change your grades in the event of a situation that they consider extreme (and they may take any incident of "cheating" as being sufficiently extreme), but you're usually okay at that point.
I know, that's three times where I emphasized the word usually. Sometimes, life offers you no absolutes, despite how much you might want one. Knowing the usual results is, sometimes, simply the best you can get.