Upvote:3
No, I don't think so.
People often misunderstand what ESTA is. It doesn't grant you permission to be in the US. It grants you permission to travel to the US (i.e. to the border) and ask for admission at the port of entry, under the Visa Waiver Program (VWP).
So in order to use this, you would have to leave the US before the expiration of your J1 status (plus grace period), then come back and ask to be admitted under the VWP. But since the VWP is intended for short-term visitors, if they ask you where you're planning to stay, and you say "in the apartment I've been renting for a year", they may not believe that you are really a visitor, and think that you are trying to immigrate. So you might be denied entry, which would undoubtedly be very awkward.