So you have already had a decision denying benefits, and you have now received a determination from the fraud unit, saying that you are determined to be overpaid $615, plus a $90 penalty because it is a fraud overpayment. So the overpayment decision is what you are wanting to appeal, specifically the $90 penalty, correct? You don't feel you were denied the benefits in error, that appeal window is already closed, but you did receive unemployment insurance benefits after giving notice and then refusing to work the hours and/or schedule during your notice period that you were offered by the employer.
Okay, you do have an appeal, that you perhaps did not understand that you were not going to be eligible if there was suitable work available for you offered by the employer which you refused. But how do you get around that question on the weekly certification, "Have you refused work, been offered any work, etc. this week?" Because from my reading, they offered you some work, but it was work that was part time, wasn't your usual schedule, or something that made it something you didn't want to do.
If you had told the unemployment system that on the weekly certification, that would have stopped the claim at that point, and they'd have done fact finding and at that point there would've been a decision made about the suitability of the work you refused, which does sound a bit unsuitable. But you didn't mention it, and that is the point at which you are believed to have committed actual fraud by lying to them. So I would say that was the false statement they are saying you made to receive benefits, it was a sin of omission, failure to admit you'd been offered work during the week you were filing for, and what will be coming up as either fraud or non fraud in the appeal of the overpayment.
Did you report that you had been offered that part time work? If not, then you pretty much didn't answer the certification question truthfully. You can say you "didn't understand' that you were supposed to report that offer of work, but the claimant is pretty much obligated to answer the certification questions correctly every week they file, and "I didn't understand that was what the question was asking about " usually doesn't work too well. But it's worth the appeal, just to pull things out longer.
Don't get in such a tizzy about the "fraud" issue. The unemployment system processes thousands and thousands of minor fraud things a year. Only the most egregious ones, the people who give them the most trouble, ever end up being prosecuted for fraud, simply because they don't have the time or resources to prosecute something this minor. They let you pay the money back, pay a penalty, and there's nothing else ever comes of it. You don't have this on a record anywhere else, its a closed agency system, you are not branded as a lawbreaker, you just got hit with a small penalty and had to pay back some unemployment insurance.