Passive mode-locking based on saturable absorbers (SAs) is an effective way to generate ultrafast pulses, while the spatiotemporal mode-locking (STML) based on SAs are rarely studied. We construct an all-fiber laser with InP quantum dots (QDs) SA, and realize multi-mode Q-switching (MMQS) and STML. In addition, by adjusting the polarization controller (PC) in the laser cavity, we obtain two different single-pulse STML states. The narrowest pulse widths of the two states are 57 ps and 32 ps, respectively, and the pulse width can be tuned continuously with increasing the pump power. By decoupling the effects of the PC and the SA in our experiment, we find that the polarization plays a key role in the selection of transverse modes. The effect of independent polarization on spatiotemporal mode-locked pulses has not been investigated before. To further analyze and understand the experimental results, we numerically solve the nonlinear Schrodinger equation, and compare the numerical simulation with the experimental results. In addition to passively adjusting cavity loss, the InP QD SA can balance the large walk off between different transverse modes, due to the spatial and spectral filtering effects. Our work has important implications for the design and application of all-fiber STML lasers.