Do fat or overweight people tend to have emotional problems more than normal weight people?

Yes. www.clinicalpsychiatrynews.com/views/fink-still-at-large... The association between obesity and severe mental illness is clear. www.ncbi.nlm.nih.gov/pubmed/21565594 obesity is associated with an approximately 25% increase in odds of mood and anxiety disorders and an approximately 25% decrease in odds of substance use disorders. www.ncbi.nlm.nih.gov/pmc/articles/pmc1913935.
Not necessarily. Being overweight does not make people prone to having more emotional problems. It does however in our society lead to the possibility of being a target of bullying which can have significant emotional harm. So the statistics may point towards higher numbers of emotional problems.