When (being) happy, dogs wag their tails.
As shown above, sometimes word "being" can be deleted.
Does the word "after" work in the same way too?
After offended, he sued the owner of the store.
Is it possible?
The conjunction "when" can easily be used with an adjective, especially one that describes a state, like "happy", "sad", "thoughtful", "distracted", etc. The meaning is "at or during the time that".
The conjunction "after" can be used with an adjective that describes a state, but in a very specific flow of events - mostly to indicate another state.
After red the fireworks turned purple, then bright yellow.
"Offended" is a state, but the sentence you have speaks of an event, not of a state in which "he" was. For all we know, he still is offended. That is why "after offended" doesn't work.