How to save drag and drop records in lightning component?

The design pattern you have isn't optimal, but for sake of keeping it simple, here's the modified drop function you'd use:

({
drop: function (component, event, helper) {
    var data = event.dataTransfer.getData("text");
    // Find the record ID by crawling up the DOM hierarchy
    var tar = event.target.closest('[id]');
    var contactData = component.get("v.ContactData");
    var index1, index2, temp;
    // Find the index of each item to move
    contactData.records.forEach((v,i)=>{if(v.Id===data) index1 = i; if(v.Id===tar.id) index2 = i;});
    if(index1<index2) {
        // Lower index to higher index; we move the lower index first, then remove it.
        contactData.records.splice(index2+1, 0, contactData.records[index1]);
        contactData.records.splice(index1, 1);
    } else {
        // Higher index to lower index; we remove the higher index, then add it to the lower index.
        temp = contactData.records.splice(index1, 1)[0];
        contactData.records.splice(index2, 0, temp);
    }
    // Trigger aura:valueChange, component will rerender
    component.set("v.ContactData", contactData);
    event.preventDefault();
}
})

There's no real error checking here, so you might want to take care of situations like when a user drops on to an area that isn't covered by a DOM element that has an ID.