It suffices to have $(a-d,b-e)$ not coprime for any $d,e$ with $\vert d\vert+\vert e\vert \le 1000$. But these are finitely many points. For each pair $(d,e)$ choose a different prime $p$ and choose $a \equiv d \pmod{p}$ and $b \equiv e \pmod{p}$ so that both coordinates are divisible by $p$ and hence the points is not coprime. These are finitely many conditions to distinct prime moduli so we can find such $a,b$ by CRT. Done. (Note that this just generalizes the classical argument that there are arbitrarily long gaps between primes.)