Does the Dilaton field change particle mass?

There appears to be some confusion here. The mass of the string spectra depends on the characteristic string length scale $ell_s$, e.g. for the first excited states of the closed string, $M^2simell_s^{-1/2}$. On the other hand, the vacuum expectation value of the constant part of the dilaton field sets the string coupling constant $g_s=e^{Phi_0}$. These two are independent parameters (not free parameters, obviously, but they do not depend on each other).

In the Einstein frame of the bosonic effective action, the gravitational coupling can be expressed in terms of the 26D Planck mass: $M_p^{-24}simell_s^{24}e^{2Phi_0}$, so the VEV of the dilaton does set the Planck mass (and equivalently the Planck length scale $ell_p=M_p^{-1}$) but this is unrelated to the mass of the string spectra, and roughly sets the scale where gravitational effects become strong. And since this VEV has no time-depenence, the question of "changing the mass" is reduced to being somewhat meaningless: change from what? The fact remains that the mass, dynamics and interactions of all the string modes are consistently derived from a single quantisation condition - so there is no notion of asking how properties change pre and post the introduction of forces or particles to augment the theory (unlike in our very modular Standard Model).

So the dilaton VEV does not change the mass of massive excitations, although it does dictate the relevance of gravitational effects at the characteristic string length scale $ell_s$, and by extension, uniquely fixes the 26D Newton's constant.

As an aside, you need to be extremely careful while invoking the "equivalence principle" in effective actions - in the bosonic case, without a mechanism to give the dilaton mass, the equivalence principle is violated. Finally, bear in mind that while constructing the low-energy effective action, the massive modes of the string are essentially "integrated out" and appear only as effective matter content - clearly the dilaton can have no bearing on the properties of these modes in this case.