Separation Anxiety

If a dog has separation anxiety this severe, the owners are probably aware of the issue and have strategies to share with you. I would ask them how they handle similar tasks in order to keep their house safe and their dog as comfortable as possible. If the dog is crate trained this may be the best solution, since many dogs find enclosed areas very reassuring, though some dogs with separation anxiety become frantic enough to injure themselves trying to escape, so it's probably not the best option for dogs who aren't already trained. For showers, you could possibly keep the dog in the bathroom with you (in a kennel or maybe tethered to the doorknob if they're likely to try to jump in the shower with you), but for many errands keeping the dog near you just isn't possible.

If the owners don't have a good system in place to prevent damage to your home, make sure the area you leave the dog is secure and as safe as possible, and be sure to have them sign an agreement that they will cover any damage the dog causes, and be extra sure they've authorized you to seek vet treatment for their dog in case the dog injures him/herself in their panic.