Basically everything is wrong, unfortunately.
$\psi^2$ isn't a mass density, it's a probability density. So there is no point in which you should be getting units of mass. As porphyrin pointed out, the units of $\psi^2$ (and hence $\psi$) will depend on the system which you are looking at.
Next, $\psi$ isn't a length amplitude. In fact, it's not an amplitude at all. When people talk about wavefunctions being amplitudes, what they are usually talking about is the coefficients of the basis states of $\psi$. For example, if you express $\psi$ as a linear combination $c_1\phi_1 + c_2\phi_2 + \cdots$, then the $c_i$'s are called amplitudes. But they aren't length amplitudes of some wave, they're probability amplitudes.
Wikipedia's pages (linked above) describe the concepts in far greater details, and you will also see these terms arise frequently when reading about QM.