仕事 and 作業: work and specific tasks
Compare work as an occupation or responsibility with particular tasks.
Work in a broad sense
仕事 (しごと) broadly refers to an occupation or assigned work. 「新しい仕事を探しています」 can mean looking for a new job.
「今日は仕事があります」 says that the speaker has work today. It does not identify the particular tasks involved.
A specific task being performed
作業 (さぎょう) refers to tasks carried out for a purpose. 「入力作業をしています」 means doing data-entry work.
「庭で作業しました」 means working in the garden. 作業 is not necessarily paid work at a workplace; it can include tasks at home.
The whole and a part
「今日の仕事は終わりましたが、家での作業が残っています」 means that today's work is finished but tasks at home remain. The scopes differ, so the statements are compatible.
「作業を中止しました」 says that a task was stopped. It does not mean that the speaker quit their job; distinguish a task from employment.
Does 「仕事を探す」 always mean looking for a lost object?
Show the answer and why
No. It usually means looking for a job or work to do. Check the object of 探す.