Do I need to update either or both the WordPress and Site URL

I have been working on my new site which was always accessible at a set IP address (xxxxx.hostcompany.com). Which is set as the WordPress and Site URL. I went to my domain name (hosted somewhere else) and update the DNS to link to the new hosting account, and it works well. When I go to companywebsite.com it loads the new site but when I click through it I get xxxxx.hostcompany.com/page-title. I would prefer if all is seen as companywebsite.com/page-title

Do I now need to change either or of the WordPress or Site URLs under settings with the website.com?

Thanks!