Mounting evidence that independent voters have soured on the Democrats is prompting a debate among party officials about what rhetorical and substantive changes are needed to halt the damage. Following serious setbacks with independents in off-year elections earlier this month, White House officials attributed the defeats to local factors and said President Barack Obama sees no need to reposition his own image or the Democratic message. Since then, however, a flurry of new polls makes clear that Democrats are facing deeper problems with independents—the swing voters who swung dramatically toward the party in 2006 and 2008 but who now are...