Removing URL and replacing with URL pointing at site with A record

Hi, I know there are previous posts relating to this but none have quite the same set of circumstances as mine. I have created a wordpress.org website and right at the start I set-up a temporary URL (may not have been the right thing to do) along with the hosting plan. The reason being the company wanted their old site to stay live while I developed the new one. Once finished I set up my clients URL as being “Parked” on the new site and they created an A Record their end to point at the new site. It’s working well but with two problems I want to overcome:
1) Once a user goes past the home page the address showing in the browser is the temporary one.
2) I need to keep paying for the temporary URL for the site to keep working.
I notice the temporary address is used a lot in the site especially for database addresses for images etc and I am wondering if there is a way I can configure the site so I don’t need the temp URL and the URL which is pointing at it shows in the browser address panel – Thanks – Richard