If your airline is a *A member you can use your miles to fly on any other airline which is also a *A member even to a destination served by your airline. (There are sometimes restrictions on the class of travel with some airlines but these are exceptions.) So you can "shop around" as you say.

If you have enough miles I suggest you consider flying in Business class. Award flights in Economy (Coach) are usually not the best use of your miles.

To check award flight availability see this thread on how to use the ANA tool.