It has nothing to do with the file. It has to do with the application you are using to view the file.
If it doesn't support color management (like e.g. Windows "Photos"), it will display according to whatever characteristics your monitor has. If that's a wide gamut model, sRGB gets oversaturated.
If you open the exported file in Photoshop again, it should display correctly if the document profile is embedded.
Color management was invented for this reason. It takes the monitor out of the equation and displays the file correctly. When the color management chain breaks, like in an application that doesn't do it, all bets are off.
Wide gamut monitors can only be used with fully color managed software. No exceptions. That's the deal you accept when purchasing one.
When you view a saved image in Photoshop, do you see the colour shift you describe? You must view your saved images using application programs that support colour management.
I hope this helps neil barstow, colourmanagement net :: adobe forum volunteer:: co-author: 'getting colour right' google me "neil barstow colourmanagement" for lots of free articles on colour management