I got cold around my mid-section and now my stomach hurts.
My stomach area got a little cold and now it's sore.
「お腹が冷える」ことを言うときの「お腹」は around one's mid-section や one's stomach area を使って言うと良いでしょう。
ーI got cold around my mid-section and now my stomach hurts.
「お腹が冷えて痛い」
ーMy stomach area got a little cold and now it's sore.
「お腹のあたりがちょっと冷えて、痛くなった」
ご参考まで！
When I was sleeping, my stomach was exposed. On top of that, the fan was blowing on me.
Because I let my stomach get cold, my stomach is extremely painful.
ご質問ありがとうございます。
"When I was sleeping, my stomach was exposed."＝「寝ているときに、お腹で出ていた。」
"On top of that, the fan was blowing on me."＝「その上に、扇風機の風が当たっていた。」
"Because I let my stomach get cold, my stomach is extremely painful."＝「お腹を冷やしてしまったので、お腹がとても痛い。」
ご参考になると幸いです。
カナダ 
アメリカ合衆国 
