How am I supposed to get past Dogamy and Dogaressa without killing them?

Like many UNDERTALE boss fights, there is a specific order that must be done in order to pass it. This one works like this.

Roll in the dirt/mud which will make them think you are a weird puppy, then Sniff to let them smell you, and then Pet. Make sure you pet both of them, and then you will be able to spare them.

Another method to spare them is to use the Stick. It's your starting weapon but becomes an inventory item if you replace it (with the Tough Glove or Toy Knife for example).

Use the stick and you will play fetch with the Dogi. This will allow you to spare them quicker and more easily, but doesn't get you their yellow text in the end credits.

This method works on all the dog bosses (Doggo, Lesser Dog, Greater Dog, Dogeressa and Dogamy).