The shortest route is from Saint John (not St. John) via Rte 7 to Oromocto, Rte 2 to St. Leonard, Rte 17 to Campbellton. Rte 2 is divided highway, Rte 7 is a major 2/3-lane highway, Rte 17 is a secondary roadway - paved 2 lane.
I hope you're going to spend some time in NB other than along these highways. The St. John River (yes, St for the river LOL!) is beautiful and the tertiary routes along both sides (Rtes 102, 845, 705, etc) are recommended if you have a smaller vehicle for touring. All these roadways are accessible via motorhome or large trailer but would be more comfortable in a smaller vehicle.
Other options for Saint John to Campbellton: 2) Rte 7 to Oromocto, Rte 8 to Miramichi, the 8 or 11 to Campbellton; 3) Rte 1 from Saint John to Petitcodiac, Rte 2 to Moncton then either Rte 126 or Rte 11 to Miramichi then as per 2.
PM me if you need more details. I live just outside of Saint John.