Rich narrative description with two stacked mimetics. どさどさ mimics the sound/motion of heavy objects being dumped or thumped down. どっかり mimics a heavy body plopping into a seat — specifically the weight-settled 'plop.' Both take the adverbial と. ~ながら 'while X-ing' marks the sigh as simultaneous background action. 腰掛ける (literally 'hip-hang') is the standard verb for sitting on a chair or ledge.