Launching WordPress with new URL

Hi, can anyone advise on how to launch a new WordPress website when portion of the URL will be changed. Our current website is hosted with company A – our new WordPress website will be hosted with company B, therefore currently our URL looks like this: “new.urlname.com” with the word “new” in front of it. When we launch the new website with company B then all the links currently point to new.urlname.com within WordPress, is there a quick URL update during launch or will WP do that automatically and remove all the links with “new.” in the beginning. I have been adding in the links using the built in WordPress insert link tool and I see WordPress adds in the full path when I do that such as “https://new.urlname.com/page”. Let me know if anyone has a quick fix to this. thank you!