We present gapless, high-resolution absorption and dispersion spectra obtained with quantum cascade laser frequency combs covering 55 cm-1. Using phase-sensitive dual comb design, the comb lines are gradually swept over 10 GHz, corresponding to the free spectral range of the laser devices, by applying a current modulation. We show that with interleaving the spectral point spacing is reduced by more than four orders of magnitude over the full spectral span of the frequency comb. The potential of this technique for high-precision gas sensing is illustrated by measuring the low pressure (107 hPa) absorption and dispersion spectra of methane spanning the range of 1170 cm-1 - 1225 cm-1 with a resolution of 0.001 cm-1.