I face the same issue when writing in French on my QWERTY keyboard.
My solution was to stop switching from EN/English to FR/French, and only use the International layout instead.
If you go in Control Panel>Change keyboards and other input methods>Change keyboards, you'll have the option to add a layout under Installed services; go to English (United Kingdom)>Keyboard>United-States-International.
The International keyboard allows you to type any accent above or under a letter.
For instance to type ä, first press shift+" and then press A.
The only drawback is off course if you want to type a quotation mark, you'll have to first press shift+" and then press space; but after some time you get really used to it.
You can also press Alt Gr (the one on the right side of the keyboard) and any other key to directly type a symbol, like if you press AltGr+L you'll get ø (also works for higher case with Shift).
If you would rather want to stay with separate keyboard layouts, you can display the Language Bar, in Control Panel>Change keyboards and other input methods>Change keyboards>Language Bar and select Docked in the taskbar; this will permanently display what layout you are using, either EN or DA, and switches automatically when you press Ctrl+Shift.
If you change the combination does this behavior still happen? If you boot into a minimal configuration does it still happen? – Ramhound – 2016-03-30T14:15:22.053
Changing to alt-shift worked.. until the latest "updates". Will try minimal configuration some time, but I think this is a "feature". – nsandersen – 2016-06-22T11:38:20.063