Thank you for asking this question that both native speakers and English learners often struggle with.
You are correct that the indefinite pronoun everybody is a singular pronoun, like the pronoun everyone, even though conceptually these pronouns represent groups of people. These singular pronouns take singular verbs, as shown in the sentences below:
Everybody needs to bring money. ("needs" is singular)
Everyone wants a new bicycle. ("wants" is singular
Although it is grammatical to use singular pronouns and possessives to refer back to these pronouns later in a sentence, most people don't. The majority of speakers of American English, except in very formal contexts, will use plural pronouns and possessives, as shown in these examples:
Everyone chooses what they want to have for dessert.
Everybody puts their own dishes in the dishwasher.
More tradition-minded speakers may stick to singular pronouns and possessives, but these can end up sounding awkward, because English has no singular, gender-neutral pronouns and possessives for people. Therefore, one has to make a choice between the feminine and masculine forms, or use both:
Everyone chooses what he or she wants to have for dessert. ("he or she" can sound wordy and awkward)
Everybody left his dirty dishes in the sink. ("his" doesn't work well if it is being applied to both men and women)