No idea why the other chose to comment rather than answer. I began to be ashamed of sitting idle. - fine, no problem. How were you sitting? I was idle. Some verbs can get adjectives to describe the state. Michael's comment would make an excellent answer to receive up-votes. . I suspect the two most common related expressions would be to sit idle to sit idly by ... while <X happens> Is this a time to sit idle? Back to work. He sat idly by as the house burned.

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My feeling is that "sit idle" and "lie idle" are generally used of objects. In the lockdown, many office computers are sitting idle. It does sound a bit strangle using it for yourself, I agree. You could say "ashamed of my idleness" or "of my failure to act", or many other things, depending on what emphasis you want.

Many verbs can be followed by adjectives. There is a whole list of "linking verbs" that do this. "Sit idle" is a widely used phrase - as is "to sit idly by while ..." https://www.macmillandictionary.com/dictionary/british/idle_1 https://blog.esllibrary.com/2014/06/05/look-appear-feel-adjective-or-adverb/ https://www.ecenglish.com/learnenglish/lessons/adverbs-adjectives-and-linking-verbs https://www.bbc.co.uk/worldservice/learningenglish/flatmates/episode72/languagepoint.shtml