Hi.
What's the form and function of the underlined part?
I was sad [that] she had to go.
And what's the form and function of the bracketed word?
Re: it is to provide emphasis [oily]
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It's an adjunct to me modifying 'sad'. But I want a better analysis, so I'm waiting for some grammarians to give a better analysis.
I was sad [that] she had to go.
The bracketed word "that" is a subordinator. It is introducing the underlined declarative content clause which is functioning as Complement of the adjective "sad".