You have a wide-gamut display. Natively, it approximately covers AdobeRGB rather than the "international standard" sRGB. Having such a monitor is, in a sense, a liability: you must set up and use colour management correctly in order to see even "standard" colours correctly. Alas, even today, not all applications do it, or set up to do it by default, in Windows. You need to examine each, as well as your Windows settings.

In practice, having a red or magenta cast (as well as oversaturated colours, which people tend to notice less) on photos viewed on a wide-gamut monitor points to an sRGB-exported image being viewed witout colour management. This naively stretches the narrow gamut of the image to the wide gamut of the display, resulting in a colour cast and oversaturation.

Alternatively, they may believe that your display is native sRGB (which is often the assumption lacking any other settings), resulting in 1:1 conversion and the same physical stretching.

Now, I'm not exactly sure what you mean by "monitor’s profile is set to...", as opposed to "“Device profile” on the computer". Do you mean a setting in the display itself? If so, the problem is likely the profile mismatch, resulting in the second scenario.

The general approach should be this.

1. Set your display to your desired mode. Ideally, it will be its native mode, whatever it's called.

2. In Windows, set the display device profile to match the mode you've chosen. Ideally, you would use a colorimeter to measure the actual result of (1) and automatically create the profile. Lacking that, you need to ensure that you are using a matching profile. If you selected the native display mode, you need to find the manufacturer-supplied profile created for that mode. The name like "ASUS PA329 ColorProfile" suggests something like that.

3. For your work, you also need to select your working profile. In a properly managed environment, this is largely independent from the above. Instead, it depends on your goals. Normally, this is one of the standard profiles such as sRGB or ProPhoto. If you only intend to share your image on FB, and generally for maximum compatibility, it is reasonable to select sRGB (thereby underutilising your display). If you have a good camera and want to enjoy its capabilities to the maximum extent your equipment allows, select a standard profile closest to your monitor (i.e. AdobeRGB in your case) or even wider.

   You select your working profile in the settings of each individual editing application (such as Lightroom).

4. Finally, you select a profile when you export/save an image. For your own archive, it often matches the working profile you've chosen at (3), so nothing extra to do. For sharing with others, you might want to convert it to the "lowest common denominator" sRGB.

   Whatever you do, you should ensure that all your saved images are tagged with the profile they are saved with, or the profile is attached to the image. (Standard profiles don't take much space). Usually there is a setting for it in the Save dialog. Theoretically, even if you don't convert the image on export, the reading app should read the tag and convert accordingly during display. Alas, this is not always happening.

For your situation, the important takeaway from (1) and (2) is that there always must be a match between what is set in the physical display settings and in the Windows device settings. And I should reiterate that a colorimeter is exactly the device that ensures that. You have a good monitor with accurate built-in LUTs, and it's a pity not to use it. For this display, a colorimeter is a small fraction of the cost.

(Note: because your display supports hardware calibration, you will need to use their (ASUS) software/methods instead of the colorimeter-supplied software. Because of this, you might get away with buying the cheapest colorimeter in a given range (e.g. Spyder 5 Express/Pro/Elite), which often differ only in the supplied software. But you need to use a colorimeter that ASUS specfies as compatible).

Without colorimeter, and without exactly knowing what is the "native" profile, one solution is to set both the device profile and the display setting to a common standard, such as sRGB. As you already discovered, this should work reasonably well. But of course, it underutilises the capabilities of your display, and still relies on a factory calibration, which degrades over time.

Finally, you need to check each relevant application (including browsers) and ensure that it supports colour management and that it is enabled. (This may be particularly problematic for video). Some applications don't use Windows settings, and for them you may need to specify the display device profile separately (and redo it every time you change display mode or recalibrate).