This is a matter of style, and you should thus be guided by your manual of style. I use The Chicago Manual of Style, which recognizes that you need to distinguish amongst words quoted in direct speech, words used to quote words, and words that need emphasis. You have considerable leeway in your choices, but here's one that helps your reader:
"I know that!" I hear you say, and then I hear a but — "but I don't
have the time", "but the kids need me", "but it's so HARD".
The direct speech is placed in double quotes. Ellipsis points aren't required for obviously incomplete quoted sentences. The first but is in italics to indicate that you're talking about the word but, which is not being used as a conjunction. The other but's are also in italics to indicate that they're emphasized words (at least they jump out to the speaker), and that bit of typography connects the first with the other three. An em dash separates the explanation from it's examples. The word HARD is capitalized to indicate emphasis that rises to the level of a shout.
You may use single quotes in place of italics to talk about 'but' as a word, but that makes things a bit busy for my taste. You may also use a colon in place of the em dash.