I'm looking for someone to work instead of me tomorrow.

--> I'm looking for someone to replace me at work tomorrow.

You should confirm the orders that came by last week and send a package.

--> You should send a package and confirm the orders that came in last week.

I order the chair on this site because this site is the cheaper than others.

--> I order the chair on this site since this site offers cheaper prices than others. 

I bought my favorite singer's goods that I couldn't buy before because it was in stock.

--> I bought my favorite singer's goods that I couldn't buy then since it was finally in stock. 

My shoes when I bought last month are shipped today.

--> My shoes, which I purchased last month, were shipped today.

He said the product will be shipped within a week but it wasn't shipped yet.

**Correct sentence! ^^ Good job!~ ^^ 

I promised to cook dinner instead of my mom.

**Correct sentence!~ ^^ Good job!~ ^^ 

He have made a junior staff send an email instead of him, but she forgot it.

--> He had a junior staff member send an email in his place, but she forgot.

I'll buy it absolute if there is in stock.

--> If it's available, I'll buy it right away.