We investigate free-running dual-comb pulses with overlapping spectra and well-controlled repetition rate difference in a single birefringent fiber cavity. Multiple linear and nonlinear soliton formation mechanisms in an all-fiber laser with partial polarization maintaining fiber are experimentally observed and validated for switchable and tunable dual-comb pulse emissions. Linear polarization mode dispersion is first exploited to emit polarization-multiplexed pulses with the upper limit of repetition rate difference at kHz level. By further tailoring linear birefringence, birefringence filter effect and nonlinear polarization evolution are well leveraged to emit hybrid mode-locked pulses with the lower limit at 10-Hz level. The lower limit of ∼12 Hz and nearly two order-of-magnitude tunable range of repetition rate difference are highlighted. Moreover, overlapping spectra and the passive mutual coherence between pulses in the free-running state are clarified, indicating the potential of the simplification of amplification system and single-cavity dual-comb source.