Can my severance benefit be stopped if I receive another job?

I work in retail management. My store is closing, and I have been offered severance benefits. I made the decision to move to Orlando and, at least at first, will be working part time at Disney World. My employer says that the agreement I am required to sign will include a clause instructing me to inform HR if I receive outside employment because severance is only meant to bridge between employers, regardless of the type of employment I find. In my research, employers will typically ask that employees not work for a competitor while receiving severance; however, is it within reason to ask that an employee not work anywhere at all? How might an employer learn of a past employee's employment status if employee's choose not to disclose?