Fully homomorphic encryption (FHE) enables the protection of data privacy at the cost of significantly higher computational demands. To alleviate its memory-bound bottlenecks, dataflow optimizations that maximize on-chip data reuse and minimize off-chip accesses could be effective. In this work, we exploit the opportunities of cross-operator dataflow optimizations in FHE accelerators. We propose a hardware-software co-design called CROPHE to overcome the inefficiencies and limitations in existing designs. On the hardware level, instead of overly-specialized functional units, CROPHE provisions a homogeneous and unified architecture that allows for flexible resource allocation and operator mapping. On the software level, the scheduling framework of CROPHE takes a comprehensive and systematic approach to explore various spatial and temporal data pipelining and sharing schemes across multiple operators, resulting in more efficient dataflow than prior work. We also propose novel cross-operator dataflow optimizations for unique FHE operators like number theoretic transforms and homomorphic rotations. The evaluation shows CROPHE significantly outperforms state-of-the-art designs by $1.77\times$ to $4.86\times$.