If your RP is stamped in your old passport it needs to transfer in your new passport, not good to gamble your job while on vacation.

First, you need to go TYPING Services, tell them about your case.Attached your new photo and bring them to any Immigration branches...You dont need to have two passport nos by going out of Doha. You can use your old passport for exit before it expire only
and if it expired while you are in vacation, you can use your renewed passport along with your old passport in re-entry to Doha, but still you need to transfer to your new passport upon arrival.