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

Yes. The association between obesity and severe mental illness is clear. 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.
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.