Let's say that Luke Skywalker has the enhancement TRUST YOUR FEELINGS. He attacks an objective and destroys it, and is focused, thus resolving that particular battle. Can he follow up by focusing Trust Your Feelings, (thus removing the focus/exhausted icon from Luke) and attack a second objective during that same turn?

Destroying an objective doesn't end an engagement ... all steps must still complete, so all units continue to strike until done (but blast damage and reward unopposed won't do anything much in such a case, they can't be used against a different objective during this engagement). Once the engagement is finished, you can start another engagement against a different objective, and yes, you can unfocus Luke using Trust Your Feelings and he can be part of that engagement if you wish because he'll be "ready".

Thanks, sir.