Condition $$\lim_{n\to\infty}\frac{\log|c_n|}{n}=-\infty$$ is sufficient for $f=0$.

Since $f(z)=e^{-cz}$ and $c_n=e^{-cn}$ satisfy all conditions, we see that this is best possible in certain sense.

This follows for example from a (much more general) theorem of N. Levinson, Gap and density theorems, AMS, 1940, page 121. Levinson's theorem allows some growth of $F$, and much more general class of sequences instead of integers.

Condition $$\lim_{n\to\infty}\frac{\log|c_n|}{n}=-\infty$$ is sufficient for $f=0$.

Since $f(z)=e^{-cz}$ and $c_n=e^{-cn}$ satisfy all conditions, we see that this is best possible in certain sense.

This follows for example from a (much more general) theorem of N. Levinson, Gap and density theorems, AMS, 1940, page 121. Levinson's theorem allows some growth of $F$, and much more general class of sequences instead of integers.

Remark. In fact Levinson generalizes a theorem of Vladimir Bernstein 1932 (Theorem 32 in Levinson's book), which also implies the result that I stated.