I will be studying in the US using a J1 exchange visa in January, but the visa expires (as well as the grace period) before my intended departure date at the end of July. The grace period expires 3 weeks before my departure date. My student accommodation is paid for up until end of July, hence why I booked the flights for this time to make the most out my paid rent. I already have an approved ESTA due to travelling to the US earlier this year, so if I change the address on my ESTA to my student accommodation, will this be sufficient while I stay the last 3 weeks in the US? I don't intend to work. I am a UK citizen.
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.