@Ceili,
There is definitely grammar, but it's a different grammar from English. Sometimes Deaf people write in ASL word order, sometimes it's just a manifestation of language problems. (As in, they haven't mastered English and the grammar issues are more about that than about an ASL influence per se.)
It's not really accurate to say that there is no "to be" verb. I can think of many ways to say it in ASL. There isn't necessarily a straight word-sign correlation, but they are two different languages. Some ASL words need multiple English words to convey the same meaning, and vice versa.
American Sign Language is mostly an amalgam of the sign language that developed on Martha's Vineyard, which had a very high percentage of deaf people (so high that most of the hearing people who lived there knew sign language, too), and French sign language that was brought over to America by Gallaudet and Le Clerc.
DrewDad, it's very sad! I've been involved in a few initiatives to try to provide services to people who have infants who failed the universal hearing screening. That was started so that deaf babies could start getting language exposure right away, instead of being finally diagnosed at age 2 or 3 or even older... but so many people just fail to follow up. Then there's the whole "fix 'em" cochlear implant mentality... sigh. (Again, some cochlear implants DO work, and I'm not absolutely anti-cochlear implant. It's just that I've seen, personally, so many situations where they DIDN'T work and the consequences were dire...)
I don't think you need to worry at all about not providing a rich ASL environment for your kids, though (as in, not talking to your wife in ASL in front of them), since they had access to a rich spoken-English environment (they're not deaf). As long as a baby/ child has access to SOME language, they can learn another language fine. The problem is when there is no real access to any language at all.