I saw this sentence, but I'm not sure what the で particle is doing in it. Everything else is fine, and I can make out what the sentence says pretty easily, but it's bugging me that I don't know specifically what the で is doing.
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When I read this question, I instantly remembered an English sentence that looked and felt very strange when I was struggling to learn English many years ago. That sentence was: "I came home sick."
My Japanese-speaking brain just kind of refused to accept it. To me, at least a word was missing in front of the last word "sick". I think I was unconsciously seeking a particle there, without which the sentence simply did not make much sense.
The same (but opposite) thing seems to be happening to you this time.
「(Phrase) + ままで～～」
is an often-used expression meaning "in the (same) situation (described by the phrase)". The state/situation is maintained. The 「で」 is the equivalent of "in" here.
"I would like to stay sad for the rest of the day."
A more literal translation would be: "I would like you/them to let me stay in the same sad situation for the rest of the day."
In highly colloquial (and sloppy) kind of speech, some people will actually drop the 「で」 just as particles in general tend to get dropped in casual speech. In writing, however, it would sound "better" if you did not drop it.