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This uses "relaxed" as a predicate adjective (or a participial adjective) describing the state or condition of "everyone" while they sat by the river. It describes how they felt.
The second option is also correct:
"Everyone sat relaxedly by the river."
This uses the adverb "relaxedly" to describe the manner in which the action of "sitting" was performed. It describes how they performed the action of sitting (e.g., they didn�t slump, they weren�t tense, etc.).