Gah! Aki-chan's Chinese dress was so cute. I almost lost control of myself — the restraints on my reason nearly came off.

Literal

Gah! Aki-[suffix-ちゃん] [gen-の] China-dress, cute-was. Dangerously, reason [gen-の] hoop [subj-が] come-off place [copula-だった].

チャイナ is shorthand for チャイナドレス 'Chinese dress / qipao.' 理性のタガが外れる 'the hoops on one's reason come off' is a vivid idiom — タガ are the bamboo/metal hoops that hold a barrel together; when they come off, the barrel (= self-control) falls apart. あぶなく 'dangerously / almost' marks the near-miss.