Sheeran will perform at Auckland's Mt Smart Stadium on March 24, 25 and 26, and at Dunedin's Forsyth Barr Stadium on March 29 and 31.
In a Facebook post, promoters Frontier Touring said the only show left with tickets remaining was the March 26 date.
They also told fans to "stay tuned" for news on additional shows.
Last week's pre-sale demand saw the pop star add two extra Auckland shows and an extra Dunedin show to his original itinerary.
Some fans reported they struggled to get tickets just two minutes after they went on sale.
The five shows will play to up to 200,000 people, breaking records as the biggest tour in New Zealand.
Frontier Touring head Brent Eccles told the Otage Daily Times Sheeran's tour was set to attract the largest crowds for a tour in New Zealand.
''It will be massive. I thought they would go well, but I'm a bit surprised about how well they've gone.''
It tops Adele's recent three-night stand at Mt Smart Stadium in March.
Tickets were being re-sold online within 30 minutes of the general sale starting.
Four tickets were advertised on resale website Viagogo for $2782.28 less than two hours after general sales opened.
Fans took to Twitter to share their ticket buying stories.
Some celebrated, while others despaired.
During last week's pre-sale, tickets for Sheeran's shows soon appeared on resale sites, often for inflated prices.
One Dunedin mother paid $1223 for three tickets which should have only cost $90 each, after her daughter, 11, bought them from Viagogo accidentally.
In a statement, Sheeran said he couldn't believe how in demand tickets to his shows were.
"I'm overwhelmed with the response, to match that record is very humbling. I really can't wait to get down there."
With 14 dates planned, Sheeran's Australasian tour is set to equal the record set by AC/DC in 2010 for the number of stadium shows played on one tour.
