During an engagement must I focus all my attacking or defending units to strike even if there are no units on the enemy side to strike against and the objective has already been destroyed?

Yes, an engagement cannot end until all units have been focused to strike.

When in doubt, check the flowchart in the back of the rule book. It shows that you "Repeat if any participating units are still ready." Thus, until all participating units have been Exhausted (whether by focusing to strike or having a focus token placed on them in some other way), you keep repeating the resolve strikes steps.

I think I have been doing tactics wrong. Lets say I have tactics and I place a token on the opposing unit who has not gone yet, does that unit have to take an additional exhaust icon for being in the battle?

Nope. Only ready (non-focused) units who are participating in the engagement are forced to strike.

So if you opponent has a non elite character committed to the force you may want to tactics another unit so that one gets double focused.

I thought so....I was doing it wrong. Thanks.