Proving $E_{DFA}$ is decidable by running $A_{DFA}$ several times
If you want to reduce $E_{DFA}$ to $A_{DFA}$, you should use the following theorem:


  If a DFA with $n$ states accepts some word, then it accepts some word of length less than $n$.


But a much more efficient algorithm would just run DFS from the initial state, and check whether some accepting state is reachable from it.