Otherwise, there are several things you can do. One is to have at least two pairs of shoes on hand, and maybe three. Rotate them, so if one ends the run wet, it has at least two days in which to dry out. Or, you can try to waterproof your feet. One way to do this is with a running shoe with a Gore-Tex liner, such as the Montrail Hurricane Ridge ($115). It's really a trail runner, but works just fine on roads.
Alternatively, waterproof your feet with Gore-Tex socks ($40 from most outdoor retailers). The advantage of these is that you can choose to use them or not at the last minute, with any pair of shoes handy. Also take care to wear fast-drying socks (no cotton) as these will help reduce friction and may even squish dry by the time you get home from a run that starts wet and ends dry.