Yes, your examples are both good. Some others: “He made a very offensive comment about her.” “Her conduct offended her parents’ sense of propriety.” “He is a repeat offender” (meaning, he has committed actual crimes — criminal “offenses” — multiples times). “I’m sorry if I offended you.”

Denise from Australia It is an offense to hit the police officer. Oh sorry, did my words offend you? The rubbish in that bin smells so bad, it is really offensive.

No offense does mean that you're not offended. I'm not sure if "Don't take offense" sounds natural to me but I would say "Don't get offended" or "I hope you didn't get offended".