new phone is showing all the contacts and settings of a phone of another person in our household

Use the following instructions to completely clear the new phone: What to do before you sell, give away, or trade in your iPhone, iPad, or iPod touch - Apple Support and then set it up again using the Apple ID of the person using the new phone.

Each person should have their own Apple ID. You can do that and still share purchases and some other things by using Family Sharing: Family Sharing - Apple

You are likely signed up with the same Apple ID. Each user should have its own.