First measurements of double-polarization observables in ω photoproduction off the proton are presented using transverse target polarization and data from the CEBAF Large Acceptance Spectrometer (CLAS) FROST experiment at Jefferson Lab. The beam-target asymmetry F has been measured using circularly polarized, tagged photons in the energy range 1200-2700 MeV, and the beam-target asymmetries H and P have been measured using linearly polarized, tagged photons in the energy range 1200-2000 MeV. These measurements significantly increase the database on polarization observables. The results are included in two partial-wave analyses and reveal significant contributions from several nucleon (N^{*}) resonances. In particular, contributions from new N^{*} resonances listed in the Review of Particle Properties are observed, which aid in reaching the goal of mapping out the nucleon resonance spectrum.