Human motion is inherently diverse and semantically rich, while also shaped by the surrounding scene. However, existing motion generation approaches fail to generate semantically diverse motion while simultaneously respecting geometric scene constraints, since constructing large-scale datasets with both rich text-motion coverage and precise scene interactions is extremely challenging. In this work, we introduce SceneAdapt, a two-stage adaptation framework that enables semantically diverse, scene-aware human motion generation from text without large-scale paired text--scene--motion data. Our key idea is to use motion inbetweening, a learnable proxy task that requires no text, as a bridge between two disjoint resources: a text-motion dataset and a scene-motion dataset. By first adapting a text-to-motion model through inbetweening and then through scene-aware inbetweening, SceneAdapt injects geometric scene constraints into text-conditioned generation while preserving semantic diversity. To enable adaptation for inbetweening, we propose a novel Context-aware Keyframing (CaKey) layer that modulates motion latents for keyframe-conditioned synthesis while preserving the original latent manifold. To further adapt the model for scene-aware inbetweening, we introduce a Scene-conditioning (SceneCo) layer that injects geometric scene information by adaptively querying local context via cross-attention. Experimental results show that SceneAdapt effectively injects scene-awareness into text-to-motion models without sacrificing semantic diversity, and we further analyze the mechanisms through which this awareness emerges. Code and models will be released. Project page: \href{https://sceneadapt.github.io/}{sceneadapt.github.io}