Hi. Lucky (fully permitted to work) textile designer here. I don't think there is any shortage of graphic designers here. If you have particular graphic skills, something that would be hard to come by here, it would be a very good idea to feature those in your CV. For instance, I know Point Carre, a graphics program specific for textile design. I checked with the NY Point Carre office and asked them for a list of businesses in the UK that used it. It turned out there were only five, unfortunately none near where I live in the UK. Still, I'm willing to fill in when their regular person is on holiday, and they will have a hard time finding someone here who knows the program. Look for businesses like the one you work for in the US, so your skills, and particularly your experience, exactly match what they need, so they have a rationalization for why they should be allowed to hire you rather than home-grown talent.
Good luck.
DJC