In nonlinear nanophotonics, cascaded second-harmonic generation (SHG) in pure plasmonic waveguides for sequential signal transformation and complex on-chip functionality remains a long-standing challenge. Precise phase matching becomes instrumental to achieve efficient SHG and enable true cascading of nonlinear processes. We experimentally demonstrate phase matching in SHG is achievable in a plasmonic system between two orthogonal modes. Accurate tuning of plasmonic two-wire transmission-line (TWTL) design parameters result in SHG from the antisymmetric excitation mode being approximately 15 times stronger than that from the symmetric excitation mode, greatly raising the conversion efficiency to 0.021%. Simultaneously, phase matching extends our TWTL operational length up to 18m. Based on the improved efficiency and operational length, we demonstrate the feasibility of cascading multiple plasmonic waveguides. We further realize a nonlinear AND logic operation using a single 18m long waveguide. These results underscore the potential of phase matched plasmonic TWTLs for compact, efficient, and scalable nonlinear optical circuitry.