Can I get a rash from my dogs dry skin?

We just got a new puppy and he has really bad dry skin all over his body. We have gave him a bath and i"ve used that shampoo before so it cant be the shampoo. Could it be from his poop or something? i have tiny red bumps all over my stomach sides chest and upper thighs and it didnt start until the day after i got the dog. I know i'm not allergic to dogs ive had then my hole life so i have no clue what it could be.