Poor puss! All our cats, from back home and my own, have been rescue cats and I've never had a problem with them scatching. They use their scratchpost and claw the carpet a little though more to keep their claws in check.
Maybe she is trying to assert a stamp on her territory, especially after being abandoned and new humans moving into her home. Could also be unhappiness and/or attention seeking. I'd have thought getting her scent about would be more territorial.
Could she have limited access to only where there is minimal damage that can be done eg wallpaper free areas? Maybe teach her that if she is destructive that her reign of the house is limited, teach her that way.
Maybe her own cosy owned space (wallpaper at will :)) as well such as a special cat den in a shed (or somewhere else warm and sheltered) with access for her? Give her some sense of territory back (though cats are notorious for picking their own and changing at will).
If it's a big problem then, as above, have her rehomed properly through a reputable organisation, maybe continue to "foster" her until a new home is found to keep space free in the local cats homes for other deserving moggies.