FPGA technology has introduced new ways to accelerate database query processing, that often result in higher performance and energy efficiency. This is thanks to the unique architecture of FPGAs using reconfigurable resources to behave like an application-specific integrated circuit upon programming. The limited amount of these resources restricts the number and type of modules that an FPGA can simultaneously support. In this paper, we propose "morphing sort-merge": a set of run-time configurable FPGA modules that achieves resource efficiency by reusing the FPGA's resources to support different pipeline-breaking database operators, namely sort, aggregation, and equi-join. The proposed modules use dynamic optimization mechanisms that adapt the implementation to the distribution of data at run-time, thus resulting in higher performance. Our benchmarks show that morphing sort-merge reaches an average speedup of 5x compared to MonetDB.