Banks of the Ohio (he leads his ex-girlfriend down the the river at knife-point, then pitches her in--perhaps in some versions he stabs her first)

The Bold Fisherman (by G.W. Hunt; the fisherman drowns in the Thames)

The Leaves of the Woodlands (by Peter Bellamy, from "The Transports": the singer plans to top herself in a river)

A Norrible Tale of the Suicidal Family (one of them drowns herself in the water-butt--the only manner of suicide in the entire song that seems remotely feasible)

Willie Drowned in Eero/Yarrow

I've read accounts of soldiers and cowboys drowning while fording rivers, but don't recall any songs specifically about this.