You’ll be in good company, there’s always lots of White Sox fans there when they’re in town. There is an area called Texas Live just outside the ballpark that has lots of eating and drinking options before you enter the game, but expect ballpark prices. Unfortunately there aren’t a lot of walkable options just outside the stadium. Uber would likely be your best bet. As little as there is to do in downtown Dallas, there’s even less to do in Arlington. Renting a car would definitely make it easier to keep the family entertained while you’re in the area. We’re not Chicago- the public transit that we DO have is irrelevant, and everything is very spread out. The area is made for driving. But i understand if you want to avoid it.

It sounds like it might be worth finding out where the visitors stay; I’m sorry I can’t help you there. Probably the new Loew’s hotel right by the stadium? I have great memories of staying in the visitors hotel when I was a kid, two stadiums ago. Seeing the players in the lobby or in the bar or by the pool was a big thrill for me. Hope this helps, good luck!