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The cases that you've read about, where people went to jail, have very different fact scenarios from yours. In those situations, the people worked the entire time they received unemployment. They made no effort to report their earnings at all and intentionally engaged in fraud in order to get both unemployment and their income.
In this situation, you are talking about one week of overlapping earnings and unemployment that you attempted to report correctly but failed. It can be considered a legitimate clerical or reporting error.
The most likely result is that they demand repayment for the overlapped week, with interest. They might consider disqualifying you for unemployment for a couple of weeks, in the future, as another punishment.
On these facts, though, this is just not the type of situation where they try to make it into a criminal
You can't do anything now but wait for unemployment to receive the independent start date and payment information from the employer. They don't want that information from you. Once they have that, they'll approach you and tell you what they have and ask for an explanation. Be honest, helpful and prepared to pay back the benefits as quickly as possible and you should be fine.