How can I find out why my washing machine trips the outlet?
The shaking probably shook a wire loose besides causing a leak, or the leak flooded over a set of wires and shorted the wire to the casing. The GFCI did its job by detecting the current to ground and shutting down the power. Reconnecting with the fault in place will cause the GFCI to trip again.

The dryer is protected by the same GFCI so when it trips the GFCI will shut down all power.

Get someone qualified to inspect it and be prepared to shell out for a new washing machine should it be irreparable.