Для начала, у вас DataGridView привязана к источнику данных или нет?

Вот минимальный пример перетаскивания строк:

using System;
using System.Collections.Generic;
using System.Data;
using System.Text;
using System.Windows.Forms;
using System.Drawing;

namespace WindowsFormsApplication1
{
    public partial class Form1 : Form
    {
        
        public Form1()
        {
            InitializeComponent();
            dataGridView1.AllowDrop = true;               
        }

        private void dataGridView1_DragOver(object sender, DragEventArgs e)
        {
            e.Effect = DragDropEffects.Move;
        }

        private void dataGridView1_DragDrop(object sender, DragEventArgs e)
        {
            Point p = dataGridView1.PointToClient(new Point(e.X, e.Y));            
            int TargetRow = dataGridView1.HitTest(p.X, p.Y).RowIndex;
            int DraggedRow = (int)e.Data.GetData(typeof(int));

            if (TargetRow < 0 || DraggedRow < 0 || TargetRow >= dataGridView1.Rows.Count-1 || 
                DraggedRow >= dataGridView1.Rows.Count-1 || TargetRow == DraggedRow) return;

            var row = dataGridView1.Rows[DraggedRow];
            dataGridView1.Rows.RemoveAt(DraggedRow);
            dataGridView1.Rows.Insert(TargetRow, row);            
        }

        private void dataGridView1_CellMouseDown(object sender, DataGridViewCellMouseEventArgs e)
        {
            if (e.Button.HasFlag(MouseButtons.Left))
            {
                int DraggedRow = e.RowIndex;
                DragDropEffects dropEffect = dataGridView1.DoDragDrop((object)DraggedRow,DragDropEffects.Move); 
            }
        }   
        
    }
    
}

Источник: https://social.msdn.microsoft.com/Forums/windows/en-US/16b0a44e-35a0-4bc8-9ccd-ec2c62c95a55/select-and-drag-a-datagridview-row-with-a-single-click?forum=winforms