You have damaged this friendship. According to what you wrote, you are a USER.
From your version it seems you think it is no big deal to ask someone to watch 3 dogs for 150 hours (you don't count the hours for this as you have for their son). As soon as she asks for a favor, you act as if she has some nerve. You act like since she is around dogs, 3 more should not be a big deal. Well she may feel since you are with kids what is one more. It is not like you are on a romantic get away. You are taking your kids to a sporting event, and the same one her kid is attending.
Her reaction from your version makes her sound immature. According to you, she said if you can't help me, you need to find a new dog watcher. For all I know her whole family, friends and therapist may have advised her to tell you off and she did just that.
To add insult to this, you act as if she is not caring 'that much' for your animals. Do you really think she does not clean up your dogs' poop for 5 days? Does she leave it in her garage, since you do not mention them being walked. I highly doubt she is as awful as you mention. You wrote here for a reason. Maybe hoping to feel you are right and she is wrong. Please, Please, Please read these comments and apologize to your friend. You are guilty of taking advantage of her.
I am writing to be helpful and not hurtful. I feel so sorry for this lady. She may have thought after all she has done for you, asking for a ride for her kid would not be a big deal.