I got something stuck in my throat and I couldn't stop coughing.
Something went down the wrong way and it made me cough like crazy.
この場合、次のような言い方ができます。
ーI got something stuck in my throat and I couldn't stop coughing.
「喉に詰まらせて、咳が止まらなくなったよ」
to get stuck in one's throat で「喉に詰まる」
ーSomething went down the wrong way and it made me cough like crazy.
「むせてすごい咳き込んでしまったよ」
Something went down the wrong way で食べ物や飲み物をむせることを言います。
ご参考まで！
カナダ 
