Two different answers while doing Inverse Z-transform
Q1: You should ask yourself if

$$\frac{z}{3z^2 - 4z + 1}\stackrel{?}{=}\frac{\frac 3 2}{z-1} - \frac{\frac 1 2 }{z-\frac 1 3}$$

really holds. You'll find out that you forgot to scale correctly.

Q2: There is no reason to consider $X(z)/z$ instead of $X(z)$ in this case.

And concerning your question about the anti-causal sequence, a multiplication by $z^{-1}$ always corresponds to replacing $n$ by $n-1$, no matter if a sequence is right-sided or left-sided.