I've used my 10-22 on FF, and while the corners are covered when zoomed in beyond ~13mm, at any aperture or zoom setting the area outside of the crop image circle is beyond a joke. Not to mention the mirror/rear element clearance issue.
If you just want a cheap FF lens, get a 28-135. Or if that's too much, buy one used, or one of the many tiny 28-80, 28-90 or 28-105 lenses.
Having said that, a white box 24-105L is the most sensible option if you can stretch to it.