The truss rod is designed to affect the relief along the entire length of the neck... from the first fret near the nut... to the last near the bridge.
There is an issue with the neck. Maybe a bad truss rod (non-functional) or a twisted neck... that's why all frets past a certain point are affected. If you've not troubleshot a similar issue in the past, seek out a well recommended luthier in your area. Also could be two or more raised frets as well, but most likely there is an issue with the neck.
The issue is with the neck relief specifically... and from your description, you need to add more relief. That's why the strings are fretting out.