I would read all of "社会人として一定の枠組みにはめ込まれれば、それなりの 窮屈さを感じ、責任を背負ってくらさなければならないと思い込んでいた" as modifying 私.
The overall structure then becomes: Rather, to me, who [X], this world was surprisingly free and put me at ease. (or something similar), where X is the modifying phrase.
The original structure in the modifying phrase doesn't say that taking responsibility would be the cause of feeling 'unease' (and I think 窮屈 here refers to social constraints/formalities).
In 社会人として一定の枠組みにはめ込まれれば、それなりの 窮屈さを感じ、責任を背負ってくらさなければならない,
社会人として一定の枠組みにはめ込まれれば is a hypothetical (if I was to be fit into the framework of 'society'... or something along those lines), and what follows would be the result (feeling constrained and having to shoulder responsibilities). That is what she had thought (思い込んでいた) at the time.