A person who wants to remarry after a sinful spouse runs off and leaves them is one thing.
Putting the official blessing on shacking-up is something entirely else again.
That's the thing... it really isn't any different. Both are sinful -- one is adultery, the other fornication. A married person is called to remain faithful regardless of what the other spouse does --- that flows from the fact that Christian marriage is indissoluble.
A marriage is presumed to be valid until and unless proven otherwise -- the fact that one spouse committed adultery and ran off is not alone sufficient to overturn that presumption. Without overturning that presumption (through the annulment process), any remarriage is presumed to be invalid and (by virtue of the couple cohabiting) adulterous.