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If this is true for every $x$ in $\prod_{\ell} \mathbb{Z}_\ell$ then it is certainly true for every $x$ in $\mathbb{Z}_p$. Now take $p=5$ and $x=2$ and you'll see that it is not possible.
I am reading a paper of Browning, Dietmann and Heath-Brown on the intersection of a quadric and a cubic. They show if the dimension is large then there are rational points provided there are smooth real points.
Nowadays such equations are treated using the primitive divisor theorem of Bilu, Hanrot and Voutier. See for example page 420 of "Number Theory: Volume I: Tools and Diophantine Equations" by Henri Cohen.
The post mathoverflow.net/questions/207024/… refers to a paper of Nagell in which the equations $x^2+x+1=y^n$ and $x^2+x+1=3 y^n$ for $n \ge 3$. You want to take $x=-p$, $y=q$, $n=\alpha$ which reduces you to the case where $\alpha=1$ or $2$.
