Though the cost of a new appliance may cost you initially, longterm newer appliances are more energy efficient and cost less to run longterm. Front loading washing machines in particular are cost efficient.
It is costlier to use more detergent than the instructions indicate on the detergent label. Most people don't realize that using more laundry detergent does not help get clothes cleaner. Today's washing machines and dishwashers use less water; using too much soap throws off the ratio of soap to water and therefore there is not enough water available to clean and rinse the clothes. The soap will leave a residue.