We study recurrence and transience for a particle that moves at constant velocity in the interior of an unbounded planar domain, with random reflections at the boundary governed by a Markov kernel producing outgoing angles from incoming angles. Our domains have a single unbounded direction and sub-linear growth. We characterize recurrence in terms of the reflection kernel and growth rate of the domain. The results are obtained by transforming the stochastic billiards model to a Markov chain on a half-strip ${\mathbb{R}}_{\mathbf{+}}\times \mathit{S}$ where S is a compact set. We develop the recurrence classification for such processes in the near-critical regime in which drifts of the ${\mathbb{R}}_{\mathbf{+}}$ component are of generalized Lamperti type, and the S component is asymptotically Markov; this extends earlier work that dealt with finite S.