Dragging-and-dropping to move a JTextArea around on JPanel

You don't really want to try and "drag" on JTextComponents, they already have functionality enabled which allows the user to click and drag to highlight text, you really don't want to be competing within this.

Instead, you want to define a "hot zone" area around the component which would allow you "highlight" the component in some and allow the user to drag the component via it.

For example...

import java.awt.BorderLayout;
import java.awt.Color;
import java.awt.Cursor;
import java.awt.Dimension;
import java.awt.EventQueue;
import java.awt.Graphics;
import java.awt.Graphics2D;
import java.awt.Rectangle;
import java.awt.event.MouseAdapter;
import java.awt.event.MouseEvent;
import java.util.ArrayList;
import java.util.List;
import javax.swing.JComponent;
import javax.swing.JFrame;
import javax.swing.JPanel;
import javax.swing.JScrollPane;
import javax.swing.JTextArea;
import javax.swing.UIManager;
import javax.swing.UnsupportedLookAndFeelException;
import javax.swing.border.EmptyBorder;

public class DragMe {

    public static void main(String[] args) {
        new DragMe();
    }

    public DragMe() {
        EventQueue.invokeLater(new Runnable() {
            @Override
            public void run() {
                try {
                    UIManager.setLookAndFeel(UIManager.getSystemLookAndFeelClassName());
                } catch (ClassNotFoundException | InstantiationException | IllegalAccessException | UnsupportedLookAndFeelException ex) {
                    ex.printStackTrace();
                }

                JTextArea ta = new JTextArea(10, 20);
                ta.setText("Bananas in pajamas");
                JScrollPane sp = new JScrollPane(ta);

                DragProxyPane proxy = new DragProxyPane(sp);
                proxy.setSize(proxy.getPreferredSize());
                proxy.setLocation(100 - proxy.getWidth() / 2, 100 - proxy.getHeight()/ 2);

                JFrame frame = new JFrame("Testing");
                frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
                frame.setContentPane(new JPanel() {
                    @Override
                    public Dimension getPreferredSize() {
                        return new Dimension(300, 300);
                    }
                });
                frame.add(proxy);
                frame.pack();
                frame.setLocationRelativeTo(null);
                frame.setVisible(true);
            }
        });
    }

    public static class DragProxyPane extends JPanel {

        public static final int BUFFER_ZONE = 10;

        private boolean mouseInHouse;
        private JComponent component;

        private List<HotZone> hotZones;

        public DragProxyPane(JComponent comp) {
            MouseAdapter ma = new MouseAdapter() {

                @Override
                public void mouseEntered(MouseEvent e) {
                    mouseInHouse = true;
                    repaint();
                }

                @Override
                public void mouseExited(MouseEvent e) {
                    mouseInHouse = false;
                    repaint();
                }

                @Override
                public void mouseMoved(MouseEvent e) {
                    Cursor cursor = Cursor.getPredefinedCursor(Cursor.MOVE_CURSOR);
                    for (HotZone hz : hotZones) {
                        if (hz.getBounds(getSize()).contains(e.getPoint())) {
                            cursor = hz.getCursor();
                            break;
                        }
                    }
                    setCursor(cursor);
                }

            };

            addMouseListener(ma);
            addMouseMotionListener(ma);

            setOpaque(false);
            setLayout(new BorderLayout());
            add(comp);

            setBorder(new EmptyBorder(BUFFER_ZONE, BUFFER_ZONE, BUFFER_ZONE, BUFFER_ZONE));

            hotZones = new ArrayList<>(8);
            // Top left, middle, right
            hotZones.add(new HotZone(0f, 0f, Cursor.getPredefinedCursor(Cursor.NW_RESIZE_CURSOR)));
            hotZones.add(new HotZone(0.5f, 0f, Cursor.getPredefinedCursor(Cursor.N_RESIZE_CURSOR)));
            hotZones.add(new HotZone(1f, 0f, Cursor.getPredefinedCursor(Cursor.NE_RESIZE_CURSOR)));
            // Left, right
            hotZones.add(new HotZone(0f, 0.5f, Cursor.getPredefinedCursor(Cursor.W_RESIZE_CURSOR)));
            hotZones.add(new HotZone(1f, 0.5f, Cursor.getPredefinedCursor(Cursor.E_RESIZE_CURSOR)));
            // Bottom left, middle, right
            hotZones.add(new HotZone(0f, 1f, Cursor.getPredefinedCursor(Cursor.SW_RESIZE_CURSOR)));
            hotZones.add(new HotZone(0.5f, 1f, Cursor.getPredefinedCursor(Cursor.S_RESIZE_CURSOR)));
            hotZones.add(new HotZone(1f, 1f, Cursor.getPredefinedCursor(Cursor.SE_RESIZE_CURSOR)));

        }


        @Override
        protected void paintComponent(Graphics g) {
            super.paintComponent(g);
            Graphics2D g2d = (Graphics2D) g.create();
            if (mouseInHouse) {
                g2d.setColor(Color.BLACK);
                for (HotZone hotZone : hotZones) {
                    g2d.draw(hotZone.getBounds(getSize()));
                }
            }
            g2d.dispose();
        }

        public class HotZone {

            private float x, y;
            private Cursor cursor;

            public HotZone(float x, float y, Cursor cursor) {
                this.x = x;
                this.y = y;
                this.cursor = cursor;
            }

            public Cursor getCursor() {
                return cursor;
            }

            public Rectangle getBounds(Dimension size) {

                return getBounds(size.width - 1, size.height - 1);

            }

            public Rectangle getBounds(int width, int height) {

                int halfBuffer = BUFFER_ZONE / 2;

                float xPos = (width * x) - halfBuffer;
                float yPos = (height * y) - halfBuffer;

                xPos = Math.min(Math.max(0, xPos), width - BUFFER_ZONE);
                yPos = Math.min(Math.max(0, yPos), height - BUFFER_ZONE);

                return new Rectangle(Math.round(xPos), Math.round(yPos), BUFFER_ZONE, BUFFER_ZONE);

            }

        }

    }

}

This sets up a simple proxy component which acts as the hot zone manager, detecting the mouse coming into or out of it and updating the cursor based on its location within in it, but it does not disrupt the normal operations of the component.

Now, this example doesn't drag, sorry, you have plenty of other examples which should be able to get you over the line, but, you could simply add a MouseListener/MouseMoitionListener to the proxy to detect when the user drags, but you will need to add some more functionality to it to determine what that drag actually means (resize or move) ;)