Yes, quickly, while you are still within the appeal window, file the appeal. DO NOT PROCRASTINATE about this. If you do not get the appeal in within the window, usually 15 days, you have no appeal, regardless of the circumstances. And this is a circumstance in which the department has actually admitted that the overpayment is their own fault. And by all means, do the appeal. It is worth it. If you had been committing fraud, knowingly falsifying your unemployment claim, there wouldn't be much appeal. But under these circumstances, definitely file it. Get on very close terms with your overpayment unit, in addition to filing the appeal. They should be quite willing to work with you.
Don't worry at all about how "professional" your appeal sounds. All unemployment insurance situations are supposed to be user friendly to the point that an attorney is not required to move you through the system if you are not very dysfunctional as in totally illiterate or something similar. Much more important that you submit something timely.
What you will need to say in the appeal letter that you send back to them is something like, "I wish to appeal the overpayment decision of xx/xx/xxxx. This overpayment, as stated in the decision, was due to agency error, not error on my part. I contacted the unemployment system when my original claim expired, as I had been instructed to do. At this time, I was signed up for, and approved for an additional claim. I filed for weeks as instructed, and was paid. I had no idea I was not supposed to be eligible for this additional claim or was receiving more money than I was entitled to, as I had been told by the unemployment system to continue making my weekly certifications, and I did so and received money for each week. I met all the requirements to receive benefits during this period, including making a job search and registering for work and reporting to the system each time I was instructed to do so."
If you are totally absolved from the overpayment, that would be peachy. I cannot predict what will happen, as I do not know all the variables.
But what will more likely happen is that you will have an overpayment which will not be forcefully collected, but will stay on your unemployment insurance account with this state (and any other state) for the rest of your working life. In other words, you will probably not be forced to pay the money back right now. You will not have your wages garnished, no forced collection process would be carried out against you. But if you ever sign up for unemployment again, you'll be charged for the overpayment, you won't be able to draw any money because the weeks you certify for will be taken by the system to pay back this overpayment.
This is not, incidentally, taxpayer money or welfare benefits we are talking about here. This is money from the unemployment insurance pool that is kept by each state to cover unemployment insurance payments. With the huge workloads of the unemployment system during the last few years, there have been many people hired to work for them who did not know exactly how to do what they were supposed to be doing. In the case of this type of double dip overpayment, the whole payment system may have misfunctioned somehow. In any case, you were not the one who would be supposed to have very superior knowledge of how the system should work. After 30 years of working with the program, I would quickly understand that I was being paid more weeks than I was entitled to. But since you were told to keep filing, and then you kept receiving money when you did file, so it appears this certainly wasn't deliberate fraud on your part.
In either case, as you say, you are still unemployed, and your financial situation is still pretty desperate. You need to request a waiver of repayment on this overpayment. Ask the unemployment fraud unit investigators (their department number will be on the notice of overpayment, Do not deal with only a basic person who answers the telephone at the unemployment insurance office, deal specifically with the fraud and overpayment unit when discussing your overpayment situation) about a waiver immediately. They understand that in the first place, this overpayment was not due to fraud, and in the second place, since you are still unemployed, you pretty much lack the ability to come up with the money to pay them back right now. Getting a waiver will require you to submit very detailed information about your finances, but when they know that you cannot repay the money, they will deal with it taking your finances into consideration.