I assume you have already completed your claim for unemployment and are waiting for word that you have been approved and that payments are coming. If so, then you would accept one of the job offers and when you get approved, you should get paid unemployment benefits through the date you return to work. Also, technically once you have a job offer (reasonable for you to take), you would no longer meet the requirements to receive unemployment benefits.
I have heard the DUA has been extremely slow. If you are awarded unemployment compensation, you will be paid from the time of the claim until you start your new job.
As long as you were qualified to receive unemployment benefits and properly followed the DUA's filing instructions each week, you will receive benefits for the period for which you were unemployed. It would not hurt your claim to call the DUA and check on the status of your benefits and to make sure that they have everything that they need from you.
This answer is provided for guidance only. DO NOT rely on it as legal advice. We DO NOT have an attorney-client relationship. You should contact an attorney in your area for a one-on-one consultation before pursuing any action or making any decisions.