I bought a car a number of years ago this old camaro, I was like 18 at the time. Just about the same situation as you.
I got the car insured, brought the temp insurance, had the title signed over, and swapped the plates from my other vehicle, brought other vehicle registration, ect.
Well comming home, friend behind me notices both rear parking lights not working. So does a cop, and he proceeded to pull me over.
He let me go though. Said to bring the car directly home and not drive it until I had it registered properly. He said I was lucky I got it insured ahead of time or he would have had to impound it. He said he “should” impound it without registration, but he didn’t.