Corollary 5.4.2 of Wall's article `Poincaré complexes I', Ann. Math.86 (1967) 213-245 gives examples of 4-dimensional Poincaré complexes $X$ with fundamental group of prime order $p\geq 23$ for which the Wall finiteness obstruction $\chi(X)$ is non-zero.
Incidentally, Theorem 1.3 of the same article is very close to the result that you attributed to me, except that I put in extra hypotheses that imply that (in Wall's notation) $\sigma(X)^*=\sigma(X)$. The thing I thought of as new in my article was considering PD groups over other rings such as $\mathbb{Q}$.
I think that Wall's $\sigma(X)$ is your $w(X)$ and Wall's $\chi(X)$ is the image of your $w(X)$ in the quotient $K_0(\mathbb{Z}[\pi])/K_0(\mathbb{Z})$.