The Dravidian languages are one of the most widely spoken language families in the world, yet there are very few annotated resources available to NLP researchers. To remedy this, we create DravMorph, a corpus annotated for morphological segmentation and part-of-speech. Additionally, we exploit novel features and higher-order models to set state-of-the-art results on these corpora on both tasks, beating techniques proposed in the literature by as much as 4 points in segmentation F1.