Awful? No. If you're worrying about betraying your cousin, she can't (and doesn't seem to want to) lay claim to every former paramour. But there are a lot of intertwined relationships here, so I would tread more carefully than if the handyman were just a casual acquaintance. First, there's the fact that he's working for you, specifically in your home. If you were to pursue a relationship with him, I'd first find another handyman and be clear with him about why.
America has the world's highest rate of children living in single-parent households, with about 40% of all births to unmarried women, which is double the rate from 40 years ago.