 Solution

Enumerate both countable models. Build an increasing sequence of finite order-preserving partial bijections. At an even stage, take the first unmatched element of the first model. Its position relative to the finitely many already matched points determines an open interval or exterior ray in the second model. Density and absence of endpoints supply an unused point there, so the map extends.

At an odd stage perform the symmetric operation starting from the second model, extending the inverse map. Every required interval has infinitely many points, so avoiding the finitely many used values is possible. The union is a total bijection, by the alternating enumeration requirements, and it preserves and reflects order because every finite stage does. Hence it is an isomorphism of structures.

This is the back-and-forth method, proving countable categoricity of the dense-order theory.