A hydraulic system is disclosed. The hydraulic system may have a unidirectional variable displacement first pump, a first actuator connected to the first pump via a closed-loop first circuit, and a first switching valve disposed between the first actuator and the first pump. The first switching valve may be configured to control a fluid flow direction through the first actuator. The hydraulic system may also have a second actuator connected to the first pump in parallel with the first actuator via the first circuit, and a second switching valve disposed between the second actuator and the first pump. The second switching valve may be configured to control a fluid flow direction through the second actuator. The hydraulic system may further have a modulation valve associated with the first circuit and configured to selectively modulate a pressure of the first circuit during actuation of the first or second switching valves.