Cross-sections and recoil polarizations for the reactions gamma + p --> K^+ + Lambda and gamma + p --> K^+ + Sigma^0 have been measured with high statistics and with good angular coverage for center-of-mass energies between 1.6 and 2.3 GeV. In the K^+Lambda channel we confirm a structure near W=1.9 GeV at backward kaon angles, but our data shows a more complex s- and u- channel resonance structure than previously seen. This structure is present at forward and backward angles but not central angles, and its position and width change with angle, indicating that more than one resonance is playing a role. Rising back-angle cross sections at higher energies and large positive polarization at backward angles are consistent with sizable s- or u-channel contributions. None of the model calculations we present can consistently explain these aspects of the data.