Boy I’d love to pry as to why your mom would get mad for not spending any money for advertising… but none of my business unless you want to share.
This is what I do for a living, help small businesses market their products and services. Craigslist is the first line offense for those who want to get going on the cheap.
Perhaps the drawings and/or photography should be handled separately from the active services such as walking, bathing, fetch, frisbee, tug-o-war. You need the main hook to stay very simple and one dimensional for people to “get it”. Your main focus is providing the activity that the owner cannot satisfy themselves.
The artwork may be best accomplished by hanging out at a dog park, and getting to know the regulars. Simply ask them if you can sketch a portrait of their animal. Most will assuredly say yes, and upon completion, they will naturally want it if it represents the animal in a noble way. That way you won’t have to juggle walking and drawing at the same time.
An easy way to make yourself look hard working is to set your rates based upon a time-frame, say 1 hour, but always work ten minutes longer so the customer sees you go the extra mile. That’s the kind of good free viral marketing that people love to brag about to their friends… “My dog walker is so great that he always works longer than promised”… It doesn’t have to be long, but just enough that they notice, unless of course they have an appointment and they need you to return promptly on time. Just let them know that you’ve got some extra time today and request to keep the animal a little longer. Perhaps then would be appropriate for doing the portrait unsolicited. “Hey I wanted to show you what I did for my portfolio of your dog…” If it’s any good, they’ll naturally want it and it would never seem that you were trying to sell them something in advance. Too easy for them to say no. Especially if they’re not an artist capable of envisioning things before they are created. You may have to show them first. Then, as word gets around, request will come for advance commissions.
@Creole Bastard knows an artist that did a portrait of a friends lost cat. Apparently that went over very well and I saw another request for one based on just seeing it. Contact him to see the painting. One of the unique aspects of it was that it included some of the cats toys and possessions in the artwork.