Copies of birth certificates are typically okay for southwest. They are used to validate the age of the child to confirm that they are flying on the correct fare, and to ensure that children are seated with an adult/in legal seats on the airplane. They want proof of age for anyone up to age 11.
As for the traveling without the second parent: while it is never really expected that you'll have a problem, it cannot hurt you to carry a notarized letter of consent to travel. There's templates for these on the internet and most banks have a notary public available. type it up, and don't have your husband sign it until in the presence of the notary.
I have only ever heard of one person having a problem, but due to the notary issue it is a really tough document to get on demand if you need it-- so better to have it and not need it than need it and be stuck in an airport somewhere.
If you've never flown Southwest through Baltimore, I would recommend going through security UPSTAIRS if at all possible. For whatever reason, the folks upstairs seem to have it more together at security.