Mode-locked oscillators with highly tunable output characteristics are desirable for a range of applications. Here, with a custom-made tunable filter, we demonstrate a carbon nanotube (CNT) mode-locked thulium fiber laser with widely tunable wavelength, spectral bandwidth, and pulse duration. The demonstrated laser's wavelength tuning range reached 300 nm (from 1733 nm to 2033 nm), which is the widest-ever that was reported for rare-earth ion doped fiber oscillators in the near-infrared. At each wavelength, the pulse duration can be regulated by changing the filter's bandwidth. For example, at ~1902 nm, the pulse duration can be adjusted from 0.9 ps to 6.4 ps (the corresponding output spectral bandwidth from 4.3 nm to 0.6 nm). Furthermore, we experimentally and numerically study the spectral evolution of the mode-locked laser in presence of a tunable filter, a topic that has not been thoroughly investigated for thulium-doped fiber lasers. The detailed dynamical change of the mode-locked spectra is presented and we observed gradual suppression of the Kelly sidebands as the filter's bandwidth is reduced. Further, using the polarization-maintaiing (PM) cavity ensures that the laser is stable and the output laser's polarization extinction ratio is measured to exceed 20 dB.