M.S.
Hello M.! I live in Rockwall, and due in 5 weeks with a little boy. My little girl is 2, so I understand your situation of needing to entertain him. Have you tried Jump and Jive? They have huge inflatable bounce houses in an indoor environment. It is a great place for him to run and play, and use some of that energy, while you could sit with the baby. Their website is www.jumpanjive.us.
Also, I sometimes take my little girl to Story Time at the Library on Wednesdays at 10. I think that this age group ends at 3, so you might want to call for the older groups' time.
I am not sure if you live in Rockwall or not, but they just came out with the new Fun Guide through the Parks and Recreation Department. You can access it online through www.rockwall.com/parks. They have all sorts of activities for children your son's age such as Art & Music Classes, Basketball, T-Ball, Soccer, etc. for really cheap. They usually last for 6 weeks and cost around $30.
Hope that information helps! I am always up for playdates too, so let me know if you ever want to meet up somewhere. Take care!
M.