If Avogadro number $N_A$, is changed from $6.022\times 10^{23}{\text{mol}}^{-1}$ to $6.022\times 10^{20}{\text{mol}}^{-1}$, this would change:

The mass of one mole of a substance is the mass of $N_A$ particles.Currently, $1$ mole of Carbon (${}^{12}\text{C}$) $=12\text{g}$ (which contains $6.022\times 10^{23}$ atoms).If $N_A$ changes to $6.022\times 10^{20}$, then $1$ mole would only contain that many atoms, and its mass would be:
$$\frac{12\text{g}}{6.022\times 10^{23}}\times 6.022\times 10^{20}=12\times 10^{-3}\text{g}$$
The stoichiometry (ratios) and definition of a gram remain the same, but the "molar mass" value changes.