Reservations agent for Southwest Airlines, leaping over policies with a single bound. faster than a speeding...
You are still able to take a carry on and your purse with you. I am sure you are aware of the pet charge to take your puppy with you. Its $50.00 each way. The puppy will need to stay in its carrier for the entire trip and the carrier must be able to fit under your seat or in the space in front of you on the cabin floor.
You may want to goto the Vet and see if there is anything they can give you to give to your puppy that will help the flight go smoothly for your puppy. Reason for this is if the puppy is constantly crying or making a commotion, the flight crew may ask that you leave your puppy behind and or you may be denied boarding.
I hope this information helps.