Both curves have no rational points.

Curves of this shape map to genus 2 curves of the form $y^2 = a x^5 + b$
(one can make $a = 1$ if one likes), by quotienting out by the group
of automorphisms generated by $(x:y:z) \mapsto (\zeta x:\zeta^{-1} y:z)$
where $\zeta$ is a primitive fifth root of unity (one can pick any two
coordinates for the scaling to act on). For curves of this type, Magma can
fairly easily compute the Mordell-Weil group (group of rational points
on the Jacobian variety). In both cases, for the first curve I tried
(using the quotient as above) the Mordell-Weil group has rank 1.
One can then use Chabauty's method (also implemented in Magma) to determine
the set of rational points on the genus 2 curve. In each case, there
are three points, but they do not lift to rational points on the given
curve.

(EDIT:) Using the quotients via the action on $x$ and $z$ leads to
curves whose Jacobians have finite Mordell-Weil groups, leading
to a simpler computation.

The three points one finds on the quotient curves correspond
to $x = 0$, $y = 0$ and $z = 0$. So if these are the only points
there, then one can easily check that there are no rational
points on the original curves.