I have this dog that I want to give away, but I don't want to tell my ex that I did it.
Is it your ex's dog as well? Why do you want to give away your dog?
No, it's only my dog. I'm just worried she would judge me. I need a break from responsibilities in my life. If I keep the dog, it'll be burdensome.
Having and caring for a dog is a big responsibility. I understand if you're no longer able to care for it. If that's the case then you should try to find it a new home that can take care of it properly, so it can lead a happy life.
Nobody has to know I got rid of the dog.
That's true. If you're not comfortable with telling others that you have to rehome the dog, then you can keep it to yourself. Just make sure you find it a good place to live.