In a large skillet over medium heat, cook the sausage, breaking it into small crumbles, until browned and fully cooked, about 7–10 minutes. Drain off excess grease.
Cut the 1-pound portion of processed melting cheese into roughly 1-inch cubes.
Lightly coat the inside of a small 2–4-quart slow cooker with nonstick cooking spray. Put the cheese cubes in the bottom of the crock.
Spoon the browned, drained sausage evenly over the cheese cubes.
Pour the entire 10-ounce can of diced tomatoes with green chilies over the cheese and sausage, including all of the liquid.
Cover and cook on LOW for 1½–2 hours, or on HIGH for about 1 hour. Stir every 30 minutes, working around the sides and across the bottom. Continue until the cheese is completely melted and the mixture is hot throughout.
Stir thoroughly to distribute the sausage, tomatoes, and chilies through the melted cheese. The dip should be creamy with no firm cheese cubes remaining. Switch the slow cooker to WARM for serving.
Serve the warm dip with sturdy tortilla chips or your preferred dippers. Use a serving spoon to transfer portions to small bowls or plates.