Dcl ships have 5 main engines and 2 electric drive motors. (Not counting the 5 thrusters) The 5 mains basically basically generate all of the electric power the ship needs for a cruise. There's a computer that dictates how much power the ship is using vs how many engines are running and need to be running. Some times all 5 are running sometimes it's 3 or less. Depends on the power load. When the mains are running, they are connected directly to a generator which creates the power. That power goes thru a power distribution box, (kind of like a big circuit breaker) which then sends it on computer command, thru the ship.
The 2 electric drive motors, are connected directly to the prop shafts, which turn the props to make the ship go. Typically both are running at any given time the ship is at sea, while the mains are alternated between the power load needs of the ship and scheduled maint. At any given time at sea, 1 main engine is dedicated for each drive motor, with the rest going to the ships power needs, as needed.
As far as the reason that you would feel a vibration on 1 side of the ship and not the other, are numerous. Take your pick. 1 prop could have a blade out of balance. Could be the way the ship is heading vs the current. A bearing set could be wearing out. Something could be wrapped around the shaft or prop itself. The isolators on the engine or drive motor could be wearing out. Take your pick.