K.H.
There are several early childhood PTAs in the area that cater to moms of children 0-5. It's a great way to meet other moms. These groups also have playgroups, mom's night out, monthly speakers, etc... and you can be as involved as you like - a little or a lot.
I've been a member of the North Dallas group (ndecpta.com), but I know there's a Preston Hollow group and a Park Cities group (University Park I think if you google it.)
Welcome to Dallas!
K.