I work full time and still dog sit. I just make sure to let prospective clients know my schedule in the first message. Most of them are fine with it since many of them work and leave their dogs at home during the day anyway. It also helps if you have a roommate or significant other who is occasionally around for company when you're not home.
I think as long as ypu are up front and honest about it you will find clients who will work with your schedule. You may get a few less clients overall but I think it's still worth it.
Make sure to mention how much love and attention they will get in between your work hours! I also like to explain how I walk and/or feed the dogs (depending on feeding schedule) right before I leave and right when I get home so they have the smallest amount of inactivity possible.
My schedule seems to work especially well with small apartment dogs who do not need a lot of exercise but I've also had larger, more energetic ones without problems.
Good luck!