What is the word for the state of being offended?

Dictionary.com lists "offendedness" as an acceptable noun:

Related forms

• offendable, adjective
• offendedly, adverb
• offendedness, noun
• offender, noun
• half-offended, adjective

But I do agree with you that it sounds a bit stilted. Regardless, it is still correct.

Though there is no synonymous noun (none that I can find, at least), there are alternatives that you can use.

Using a gerund, you can convert a verb into a noun. This means that "to take offense" and "to be offended" are valid when you use them as a gerund:

• I tried to avoid taking offense.
• I know I am sensitive to being offended.