He seems pretty occupied with all his household chores.
この場合、次のような言い方ができますよ。
ーHe seems busy doing his housework.
「彼は忙しいそうに家事をしています」
to seem busy ~ing で「〜をして忙しそうだ」
ーHe seems pretty occupied with all his household chores.
「彼は家事で結構忙しそうだ」
to seem occupied with ... で「... で忙しそうにする」
ご参考まで!