Here is the situation. I attack with a At-At and they defend with only a X-wing escort. I win the edge battle. The At-At has to strike, but do I have shot down the X-wing escort? Can i choose not to use the blaster icons or not target the X-wing escort with them?

Striking is not optional. Page 20:

5. Resolve Strikes

A player does not have the option to pass during this
step. If he has at least one ready participating unit, it
must focus to strike.

&

resolving a strike
To resolve a strike, a player follows these steps in order.
1. He chooses one of his participating ready units.
2. He focuses the unit to strike (i.e. places a focus
token on the card).
3. He resolves the striking unit’s combat icons by type.

If the only unit damage icons were white then you could choose not to win the edge battle, other than that if the unit damage is black you have no choice,