On the factorization of iterates of $x^d+c$ in large degree

Let $K$ be a function field of a curve in characteristic zero or a number field over which the $abc$-conjecture holds, fix $\alpha \in K$, and let $f_{d,c}(x)=x^d+c$ for some $d\ge 2$ and some $c\in K$. Then for many $c$ and $d$, we prove that $f_{d,c}^n(x)-\alpha$ has at most $d$ factors in $K[x]$ for all $n\ge 1$. For example, when $\alpha =0$ we prove that the set $$\{d:f_{d,c}^n(x)\text{has at most }d\text{ factors in }K[x]\text{ for all }n\ge 1\text{ and all }h(c)>0\}$$ has positive asymptotic density. We then apply this result to compute the density of prime divisors in certain forward orbits and to establish the finiteness of integral points in certain backward orbits.