You're rapidly making a mess out of this. Stop, sit back, and stop thinking there's a "problem" to "fix". There isn't - you just need to use software that is color managed. That disqualifies IE right off. Stop using it, throw it away. It's useless with wide gamut displays. Use Firefox, which has proper color management.
OK. Save For Web in sRGB, embed profile. So far so good. But:
Don't ! set your working space to Monitor Color!. That turns off display color management which is the very last thing you want with a wide gamut monitor. You could sort of get away with that with a standard gamut monitor, because it's not all that different from sRGB anyway. So you wouldn't notice the difference (but it's there). The fact that your Adobe RGB files look right in Photoshop is purely coincidental. Any other profile will look wrong.
With a wide gamut display you absolutely and unconditionally need a fully color managed pipeline. That means 1. an embedded document profile, 2. a valid display profile (Spectraview or other calibrator), and 3. an application that reads both profiles and does the conversion from one to the other as the image is sent to the display.
See, it's not just the document profile. That's half of it. The other half is the display profile. IE doesn't use the display profile, instead substituting sRGB. And that's very wrong with that monitor. Firefox is fully color managed if there is an embedded document profile. But it can be configured to color manage even if the image is untagged (and a lot of material on the web is untagged). It does this by assigning sRGB to the image.
To configure this - and you really need that with a wide gamut monitor - type "about:config" without the quotes in the address bar and hit reload. Scroll down to gfx.color_management.mode, and change it from 2 to 1. Relaunch. All web material will now appear correctly regardless.