Learn Turkish: Çiçekleri damıtarak esans yapmadan edemiyoruz. - We cannot help but make essence by distilling the flowers.

This is a common structure for 'cannot help but do' or 'can't stop oneself from doing' something. It pairs a verb with the negative ending -madan (without doing) and edemiyoruz (we are unable to do / manage). Literally, it translates to 'we cannot manage without making'.

Could we use duramıyoruz instead of edemiyoruz here?

Yes, yapmadan duramıyoruz would mean exactly the same thing. Edememek and duramamak are virtually interchangeable when paired with -madan to express a strong psychological compulsion or inability to resist doing something.

What role is the -arak suffix playing in damıtarak?

The -arak suffix forms an adverb that answers the question 'how?'. Here, it attaches to damıtmak (to distill) to mean 'by distilling.' It tells us exactly how the essence is made, connecting the two verbs.

Why does Çiçekleri take the accusative case suffix -i while esans does not?

Çiçekleri is a specific, definite object ('the flowers'), so it needs the accusative case. Esans refers to the essence in a general, indefinite sense (just 'essence', not 'the essence'), so it stays in its bare dictionary form.