This paper presents a design approach for dual-band dual-polarized (DBDP) waveguide antenna array. A novel transmission network using orthogonal parallel plate waveguide (PPW) is proposed for the excitation of dual polarization. Dual-band bandstop structures are incorporated into the PPW feeding network. They suppress the leakage between the two polarizations and improve the transmission efficiency of the network. They also enable a wide range of frequency ratio from 1 to 2.64 between the two bands. A new orthogonal linear signal generator (LSG) enables ultra-wideband, dual-polarized quasi-TEM excitation, ensuring consistent and reliable performance across a broad bandwidth. Fed by four sequentially rotated single-ridge 1-to-4 power dividers, the LSG effectively generates dual-polarized signals. For demonstration, a K/Ka-band shared-aperture dual-polarized array antenna has been designed, analyzed and experimentally verified. The array features a wide impedance bandwidth (17.7-21.2 GHz and 27.5-31 GHz), high isolation (>50 dB), and excellent cross polarization discrimination (XPD) (>40 dB). The prototype demonstrates high aperture efficiency up to 80% and consistent radiation patterns at both bands, affirming its potential for SATCOM systems.