Actually, when the new entitly takes over, you will have been laid off from the old job. You will be leaving one company, and if they offer you a job with the new entity, it will be your decision to accept the new job with the new company, with new W-2s and new employer, new job and move from one to another without going through the unemployment process.
If they do not offer you a job with the new contractor, or if the job they offer you is signigicantly different in pay or description or some other important aspect from the old job, you have the option not to accept it, and to file for unemployment benefits. You will tell them about the job offer from the new company, and a decision will be made whether you have been offered suitable work.
Of course there are lots of things to consider. In the first place, the company, being a non-profit, may or may not be covered by the unnemployment system. There may not be a lot of wages in the accounty to set up a claim. The amount you make drawing unemployment is probably not going to be as much as the amount you'd make drawing benefits, and will end quickly
But actually, you do have a bit of leeway. Many new companies do not want the old companies employees. These employees are legitimately laid off. Sometimes companies do not offer as good a job, and the employees of the old company will have to make a decision about whether to accept or not accept. Usually it's a good move to go with the new job if offered and try to find another job soon instead of fighting the issue out in unemployment. But they don't buy the old employees when they take over a contract. neither party is under a mandate to take the other job or to hire the other company's employees. It's just what your options will be that are up for discussion.