This is what happens when color management stops, in applications that don't do it. Photoshop is right and the others wrong. There's no problem here, you're just not taking the necessary precautions.
Short version: Always convert to sRGB files that are to be viewed in non-color managed applications.
Long version: When Photoshop displays an image, it converts from the document profile to the display profile created by the Huey. Non-managed applications don't do that, and just send the raw RGB numbers in the file directly to the display unmodified.
Most monitors have a native response very close to sRGB. This means that an sRGB file will display roughly correctly on such a display, even without color management.
You should understand that calibration is two separate processes: one is a basic correction of the display - that's the calibration part - and the other is the creation of a display profile, which is a full and detailed description of the display's response, with much higher precision. This is what Photoshop and other color managed applications use. So when you run the Huey, the basic correction applies globally. But the profile doesn't.
So why is it after I convert, color correct, make a jpg and burn a DVD is that final image not like the one I see in Photoshop? All that work is out the window. I've never had this problem before. I do have it set to sRGB but I have to fix this somehow.
It could be that the Huey display profile has gone bad. That happens. That would only affect Photoshop and other color managed apps, not the applications that don't use the profile (like Windows Explorer and Picasa).
Try to set sRGB as default display profile in Control Panel > Color Management > Devices, and relaunch Photoshop so that it can pick up the new profile: