I am looking for a word or phrase to describe someone who thinks their own problems should supersede everyone else's, and who is childish about it, too boot.
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Specifically regarding females, specifically in East Asia, Princess sickness refers to exactly the personality traits you describe in the question.
Princess sickness, alternatively known as princess syndrome or princess disease [...] is a neologistic term used colloquially in some parts of Asia, notably Hong Kong, Macau, mainland China, South Korea, Japan and Taiwan, to describe a condition of narcissism, egocentrism and materialism in women, or "princess" behaviour. Conversely but less commonly, men with a similar outlook may be described as having "prince" sickness. - Wikipedia