We have an older home with counters/cabinets that are only 23" deep. Are there any built-in dishwashers out there that will fit such a space (i.e. less than 24" deep)?

Yes, there are. European dishwashers often tend to be a little narrower and shallower than U.S. models. I know for certain that Bosch, Thermador, Miele and Asko all offer models about 22.5 inches deep. Knowledgeable employees at a good appliance store or retailer should be able to help you find the right model for you--just don't expect it to be under $500.