First, fonts are drawn around the "base line", that is, the y position represents the baseline, so the text can grow above it and below it...

So, instead of using -textheight, you should be using FontMetrics#getAscent
Second, you need to keep track of all the previous locations text was painted, you could do this by using FontMetrics#getStringBounds, which returns a Rectangle2D and simply keep a list of this, which you can iterate over to check if the new text intersects any other, as an example...

import java.awt.Color;
import java.awt.Dimension;
import java.awt.EventQueue;
import java.awt.Font;
import java.awt.FontMetrics;
import java.awt.Graphics;
import java.awt.Graphics2D;
import java.awt.geom.Rectangle2D;
import java.awt.image.BufferedImage;
import java.io.BufferedReader;
import java.io.File;
import java.io.FileReader;
import java.io.IOException;
import java.util.ArrayList;
import java.util.Collections;
import java.util.List;
import java.util.Random;
import javax.swing.JFrame;
import javax.swing.JPanel;
import javax.swing.UIManager;
import javax.swing.UnsupportedLookAndFeelException;
public class Test {
    public static void main(String[] args) {
        new Test();
    }
    public Test() {
        EventQueue.invokeLater(new Runnable() {
            @Override
            public void run() {
                try {
                    UIManager.setLookAndFeel(UIManager.getSystemLookAndFeelClassName());
                } catch (ClassNotFoundException | InstantiationException | IllegalAccessException | UnsupportedLookAndFeelException ex) {
                    ex.printStackTrace();
                }
                JFrame frame = new JFrame("Testing");
                frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
                frame.add(new TestPane());
                frame.pack();
                frame.setLocationRelativeTo(null);
                frame.setVisible(true);
            }
        });
    }
    public class TestPane extends JPanel {
        private BufferedImage img;
        public TestPane() {
            img = new BufferedImage(1200, 650, BufferedImage.TYPE_INT_ARGB);
            try (BufferedReader br = new BufferedReader(new FileReader(new File("Words.txt")))) {
                List<String> words = new ArrayList<>(25);
                String text = null;
                System.out.println("Read");
                while ((text = br.readLine()) != null) {
                    words.add(text);
                }
                System.out.println("Loaded " + words.size());
                Collections.sort(words);
                Random rnd = new Random();
                Font font = getFont();
                Graphics2D g2d = img.createGraphics();
                g2d.setColor(Color.WHITE);
                g2d.fillRect(0, 0, 1200, 650);
                List<Rectangle2D> used = new ArrayList<>(25);
                for (String word : words) {
                    int size = rnd.nextInt(37) + 11;
                    Font drawFont = font.deriveFont((float) size);
                    FontMetrics fm = g2d.getFontMetrics(drawFont);
                    Rectangle2D bounds = fm.getStringBounds(word, g2d);
                    System.out.println("Positing " + word);
                    do {
                        int xPos = rnd.nextInt(1200 - (int)bounds.getWidth());
                        int yPos = rnd.nextInt(650 - (int)bounds.getHeight());
                        bounds.setFrame(xPos, yPos, bounds.getWidth(), bounds.getHeight());
                    } while (collision(used, bounds));
                    used.add(bounds);
                    g2d.setFont(drawFont);
                    g2d.setColor(Color.BLACK);
                    g2d.drawString(word, (float)bounds.getX(), (float)bounds.getY() + fm.getAscent());
                    g2d.setColor(Color.RED);
                    g2d.draw(bounds);
                }
                g2d.dispose();
            } catch (IOException exp) {
                exp.printStackTrace();
            }
        }
        @Override
        public Dimension getPreferredSize() {
            return new Dimension(1200, 650);
        }
        @Override
        protected void paintComponent(Graphics g) {
            super.paintComponent(g);
            Graphics2D g2d = (Graphics2D) g.create();
            g2d.drawImage(img, 0, 0, this);
            g2d.dispose();
        }
        protected boolean collision(List<Rectangle2D> used, Rectangle2D bounds) {
            boolean collides = false;
            for (Rectangle2D check : used) {
                if (bounds.intersects(check)) {
                    collides = true;
                    break;
                }
            }
            return collides;
        }
    }
}
The red rectangles are just for demonstration purposes and you can get rid of those.
This does, however, demonstrate a small problem, not all the text fills all of the rectangle.
A more complex solution would be to use a TextLayout to generate a Shape of the text, which would allow words to be grouped much closed together.
Have a look at Assigning a image to a String for a demonstration of how this can be generated