Your diagnosis of a bad solonoid may not be complete. If you have a bad connection (do you really mean open circuit or short circuit)it must be corrected. If the wire is broken where it enters the solonoid body (open circuit) and cannot be fixed then a new or used solonoid should cure the problem.
Can you explain the circumstances when you measured 11.7 Volts, was the red OD light on at the time? That voltage would indicate that the solonoid was energized.