Hi Sufyan T.
Since your site can’t be both updated and not updated, like schrodinger’s cat, the domain that is showing old content is coming from a cache somewhere, browser, access provider, CDN… hard to say. So let’s start with your browser:
https://en.support.wordpress.com/browser-issues/
Do other visitors in other browsers have the same issue?
Based on what you are saying it is hard to tell. What kind of content is it? I assume the domain you don’t edit with is not updating? If the browser issues page does not help, give us some nice troubleshooting data to work with.
— Sheryl
Also, using two domains is discouraged by search engines. If two domains are called for, you might consider having one forward to the other, so you, search engines, and your visitors are always browsing the site with one domain.