How do I disable the monthly event movement mode and keep the interaction in weekly?

You only have to verify that you are not in the monthly view for when the calendar receives the drop event, you revert the operation, this is what you must put in the data passed to the fullCalendar function that you have to add

eventDrop: function(ev,delta,revertFunc){
    //Aquí verifico que no esté seleccionado el botón "Month"
    if(!$('.fc-month-button').hasClass('fc-state-active')){
        //revierto la operación, esta función no hay que declararla es el 3er parámetro pasado al evento drop
        revertFunc(event);
    }
}

Here I leave the full code of my fullcalendar

$('#calendar').fullCalendar({
    header: {
        left: 'prev,next today',
        center: 'title',
        right: 'month,agendaWeek,agendaDay'
    },
    defaultDate: '2016-01-12',
    editable: true,
    eventLimit: true, // allow "more" link when too many events
    events: [
        {
            title: 'All Day Event',
            start: '2016-01-01'
        },
        {
            title: 'Long Event',
            start: '2016-01-07',
            end: '2016-01-10'
        },
        {
            id: 999,
            title: 'Repeating Event',
            start: '2016-01-09T16:00:00'
        },
        {
            id: 999,
            title: 'Repeating Event',
            start: '2016-01-16T16:00:00'
        },
        {
            title: 'Conference',
            start: '2016-01-11',
            end: '2016-01-13'
        },
        {
            title: 'Meeting',
            start: '2016-01-12T10:30:00',
            end: '2016-01-12T12:30:00'
        },
        {
            title: 'Lunch',
            start: '2016-01-12T12:00:00'
        },
        {
            title: 'Meeting',
            start: '2016-01-12T14:30:00'
        },
        {
            title: 'Happy Hour',
            start: '2016-01-12T17:30:00'
        },
        {
            title: 'Dinner',
            start: '2016-01-12T20:00:00'
        },
        {
            title: 'Birthday Party',
            start: '2016-01-13T07:00:00'
        },
        {
            title: 'Click for Google',
            url: 'http://google.com/',
            start: '2016-01-28'
        }
    ],
    eventDrop: function(ev,delta,revertFunc){
      if(!$('.fc-month-button').hasClass('fc-state-active')){
        revertFunc(event);
      }
    }
});