I encountered similar issues with my Lulzbot Mini with ABS. (Oddly enough, the white ABS was far worse than the black or silver ABS from the same manufacturer.)
The problems pretty much went away after adding an enclosure and adjusting the first layer settings.
The enclosure solved the problem of the corners of large parts lifting off the bed, the problem being that ABS shrinks as it cools, and enclosing it keeps the part warmer and away from drafts and other nasties. I started by trying a simple hacked-up enclosure made from cardboard boxes and masking tape. The results were dramatic enough that I investing in the “Printed Solid” enclosure (it’s nice to be able to see the entire printer and print bed, and worth a few dollars more to me).
Adjusting the first layer was largely trial-and-error, and seemed to impact small parts more than the larger ones – which makes sense, since the large ones have more contact with the bed to begin with. In a nutshell, over-extruding that first layer is the key. With the original (aka “legacy”) Cura, which I still use since I find Cura2 to be far, far too buggy at present, the first layer width needs to be 125%, and the first layer height I usually set to 0.30 or 0.35 – the idea I’m working with is that the head needs to be close enough to “squash” that over-extruded first layer onto the PEI surface, but not so close that it impedes the flow. As mentioned, the white ABS was just difficult – so using a raft has been necessary for some of the really small parts. And raising the print temperature by 5C for the white ABS was helpful not just for the first layer, but also in making the layers adhere better in general.