High-statistics cross sections and recoil polarizations for the reactions γ+p→K++Λ and γ+p→K++Σ0 have been measured at CLAS for center-of-mass energies between 1.6 and 2.3GeV. In the K+Λ channel we confirm a resonance-like structure near W=1.9GeV at backward kaon angles. Our data show more complex s- and u- channel behavior than previously seen, since structure is also present at forward angles, but not at central angles. The position and width change with angle, indicating that more than one resonance is playing a role. Large positive Λ polarization at backward angles, which is also energy dependent, is consistent with sizable s- or u-channel contributions. Presently available model calculations cannot explain these aspects of the data.