In this work, we demonstrate a comprehensive study on the nonlinear parameters of carbon nanotube (CNT) saturable absorbers (SA) as a function of the nanotube film thickness. We have fabricated a set of four saturable absorbers with different CNT thickness, ranging from 50 to 200 nm. The CNTs were fabricated via a vacuum filtration technique and deposited on fiber connector end facets. Each SA was characterized in terms of nonlinear transmittance (i.e. optical modulation depth) and tested in a Thulium-doped fiber laser. We show, that increasing the thickness of the CNT layer significantly increases the modulation depth (up to 17.3% with 200 nm thick layer), which strongly influences the central wavelength of the laser, but moderately affects the pulse duration. It means, that choosing the SA with defined CNT thickness might be an efficient method for wavelength-tuning of the laser, without degrading the pulse duration. In our setup, the best performance in terms of bandwidth and pulse duration (8.5 nm and 501 fs, respectively) were obtained with 100 nm thick CNT layer. This is also, to our knowledge, the first demonstration of a fully polarization-maintaining mode-locked Tm-doped laser based on CNT saturable absorber.