All colleagues from the two existing stores will remain with the business, with Hartlepool manager Mark Slater set to head up the new-look team of seven.
The shop, which will have five customer desks and will offer a full foreign exchange service, will open its doors on Friday 18 September in the town's M Parkway Shopping Centre.
The shopping centre posted a picture of a window poster advertising the new store last week with the caption: "Coming soon to M Parkway."
David Chambers, Dawson & Sanderson's Managing Director, said the decision to combine the two branches came in response to changing shopping habits.
"We have been looking for the right opportunity in Teesside for a number of years," he said. "Our stores in both Middlesbrough and Hartlepool have increasingly found themselves out of position as shopping habits and town-centre footfall have changed.
"Coulby Newham has always been on our radar, but there hasn't been a suitable unit available. When we became aware of this opportunity, we felt we had to take it."
'We're investing in Teesside'
Chambers said the new shop, which will be one of Dawson & Sanderson's largest by footprint, would benefit from "strong, consistent, year-round footfall" in the shopping centre, which is adjacent to a Tesco superstore offering free parking and "better accessibility for customers from across Teesside".
The agency will have 19 branches once Middlesbrough and Hartlepool close and Coulby Newham is up and running.
Chambers added clients would benefit from the combined knowledge and experience of both teams, who will be "in place and ready" for the opening in a couple of weeks time.
"Rather than reducing our presence in Teesside, we see this as an investment in the area, bringing two experienced teams together in a larger, better positioned store that we believe has significant potential," he said.
Chambers said Dawson & Sanderson had no immediate plans to grow its shop network in the coming months. "Our focus is on continuing to strengthen our existing network and ensuring our stores are in the right locations for the future," he said.
"The Coulby Newham relocation is a good example of that strategy, investing where we believe there is an opportunity to create a stronger and more sustainable presence."