If she perceives you as just "sitting around all day at the computer doing nothing," it might help to make, and keep to, a strict schedule, which you mention to her and perhaps even keep posted where she can see it. I mean, if you were going off to campus for a class you would probably have told her about that and even given her a class schedule, right? Same if you were working, say, a part-time job. So think of your online class and your job-hunting like that. And then let her know when you're available to do chores.
"Mom, from 9am to 11am M thru F I will be working on my class, which is online. From 11am until 2pm, excluding lunch, I will be researching and applying for jobs online. After 2pm every day, and all day on the weekends, I'll be available to do chores."
Then you can say, "Mom, I told you, I'm working on my class right now" or "Mom, I told you, I'm applying for jobs right now." Treat the times you gave her like a work or class schedule that really can't be interrupted or postponed. And then when it's time to do chores, get up and do those promptly, so you're not the kid who has to be nagged into working.
Maybe ask her for a list of things she wants done regularly in advance, so you don't have to check with her all the time and you can plan them. I think if you're living at home rent-free, doing chores is perfectly reasonable, but there should be room for negotiation in the actual chores, when they're done, and how many you do. Maybe if you have a set schedule and a set list of chores, both of you will feel like you're able to make more progress, rather than just doing whatever she wants whenever she thinks of it.
I understand that your computer work is personal to you, but if it would help her to see the website for your class, you might try that. Or perhaps challenge her to sit down at a company's website and figure out how to find and apply for a job with them! I myself had no idea how complicated and extensive the process could be until a colleague was job-hunting and telling me about it.