I sold my P3 once i got my P4P is there a way that the new owner of P3 can register it in his name now instead of it remaining in my name?

Nope. From what I've read the name you give it as the beginning is the name is stays with since that's the name its registered the warranty under. I may be wrong but I'm fairly confident it cannot be changed.

Okay thanks. So my next question is who would be at fault/liable if any kind of accident happened while the new owner is flying it; the new owner since he was the pilot or the person whom the drone is registered to?

The info you used when activating the drone cannot be changed. While the buyer can register a new DJI account for himself and login to the APP with his own account.
The flight records and data will record the time for flying, if any problems occur when he flies the drone, please feel free to contact us, we'll help him deal with the case.

The person you sold the drone to will have to register with the sUAS  Registration with the FAA in order to fly here in the USA. The FAA will issue a FAA # to be placed on the drone before flying. Then that person with the FAA # will be responable for flying there drone. The links below should help.

https://www.faa.gov/uas/getting_started/
https://registermyuas.faa.gov/

I would always get a bill of sale that specifies serial and model number, date and time, and of course new owner info. Signed by both parties and maybe even notarized to protect yourself.

Ok thanks for the good advice