|Hi Jessica -
Yup - pets are a perfect tail-wagging vehicle for bringing in that dreaded urushiol oil. Poison oak, ivy, and sumac all have urushiol oil but vary slightly in composition with roughly same effect as all cause allergic contact dermatitis. This is an immune system reaction to what is viewed as a "foreign body" and the reaction will vary from person to person and from time to time. It's hard to say what's normal except for some of the obvious (streaky, raised, itchy rash). Keeping the rash clean and dry as you are doing is your best bet to reducing the duration. Seeing the doctor when you feel that things aren't going right or self-treatment isn't working is a good idea.
Having a rash all summer? Yup been there. Recontamination is not that uncommon. If I suspect that my dog has been running through poison ivy, I either hose him down or send him swimming. This seems to help. Keeping the dog off the furniture isn't a bad idea either but don't forget those hugs - I can generally tell how I got poison ivy based on where and when it breaks out. Inside the arms? Probably bear-hugged the dog.
Find a treatment that works for you and keep it handy. Welcome to the club of fellow sufferers :-)