In this paper, we experimentally investigate the thermoacoustic instability issue in an annular combustor with 16 oblique-injecting premixed swirling burners. It is demonstrated that there exist three dominant modes in a narrow operating range: a Helmholtz mode, a first-order azimuthal mode, and a second-order azimuthal mode. Their modal frequencies are consistent with the simulating prediction of a Helmholtz solver. Our present investigations are more focused on the second-order azimuthal modes which are comparatively infrequently observed in the experiments of model annular combustors. The dynamic mode decomposition approach is used to postprocess the high-speed flame images, revealing the primary dynamic structure of the flame responses for the three self-excited thermoacoustic modes. A pressure field analyzing ansatz has been involved to feature the self-excited azimuthal instabilities, including their dynamical nature (standing, spinning, or mixed) and the time-varying pressure antinodes. Results indicate that the first-order and second-order azimuthal modes both exhibit a standing nature with relatively fixed pressure antinodes. Additionally, in a transition case where these two azimuthal modes co-exist, the first-order azimuthal mode behaves as a weakly oscillating standing mode whose pressure antinodes exhibit a fat-tailed distribution. Exceptionally, the second-order azimuthal mode is split into a pair of nondegenerate modes with two close frequencies. And the split pairs are found to yield distinct pressure antinodes that are orthogonal to each other.