You might try to contact your state unemployment office again to see if they can tell you who issued the (PUA )1099-G. Then you can contact that agency to see if they can get you a duplicate 1099-G. Someone issued it, so someone has a copy of it. You just have to find out who that is to get a duplicate.
If you received your unemployment from the State Unemployment Office, then they have to be able to either help you get a copy of the 1099-G or tell you where to go to get one.
You are supposed to include all income when you file your income tax return.
Did you keep records of how much you received? If so, you should be able to include it in your return by saying you didn't get a 1099G so you can report it when you file your return.