The profile, in this case Adobe RGB, follows the file, that's the whole idea. The profile defines the colors you get with those specific RGB numbers. The only option when you save is to not embed it, which is usually a very bad idea, because then the RGB numbers no longer have a specific meaning. They're just arbitrary numbers.
The working space mainly matters when you create a new file. The document profile, if it already has one, overrides the working space. Put it this way: each file has its own working space, and then you have a backup working space in case the file has none.
(You can change this behavior in color settings > color management policies, but don't. The default is "preserve embedded profiles", and that's the only sensible setting. The other two will quickly send you out swimming with no dry land in sight. So to speak).
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All that said, working in Adobe RGB is probably not a good idea until you get a better understanding of the implications. sRGB is the safe choice, it will usually display correctly without any special considerations. So if you have an Adobe RGB file and you want to use the safer sRGB instead, you convert to sRGB, with the menu item Edit > Convert to Profile.
Don't be confused by the fact that you then have sRGB as both document profile and monitor profile. That's coincidental, and just to set up a safe workflow. These two profiles are still separate and serve different purposes. The document profile defines the numbers in the file; the monitor profile describes the behavior of the monitor. But they're similar enough so that you can use it in both places.
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So. What do you see now, with sRGB as monitor profile? You must close down Photoshop and relaunch it for the new profile to take effect. Are they similar or still different?
