We demonstrate generation of high-purity photon pairs at 1560 nm in a single spatial mode from a periodically-poled KTiOPO4 (PPKTP) waveguide. With nearly lossless spectral filtering, the PPKTP waveguide source shows approximately 80 % single-mode fiber coupling efficiency and is well suited for high-dimensional time-energy entanglement-based quantum key distribution. Using high-count-rate self-differencing InGaAs single-photon avalanche photodiodes configured with either square or sinusoidal gating, we achieve > 1 Mbit/s raw key generation with 3 bits-per-photon encoding, and, to the best of our knowledge, the highest reported Franson quantum-interference visibility of 98.2 % without subtraction of accidental coincidences.