C.E.
Mansfield definitely has year-round swimming, but I don't think they have a summer team. What you want is a team aligned with TAAF (Texas Amateur Athletic Federation.) The Keller Barracudas is one such team. I know they are too far from you, but you may be able to contact them and find out if there is still an Arlington team. It used to be the Arlington Stingrays and they swam out of the Hugh Smith pool, but they aren't around anymore. I know there are teams in Grapevine, Southlake and Bedford. Call the directors of Arlington, Mansfield or Grand Prarie Parks & Rec, the Mansfield Natatorium or email the person on TAAF in charge of swimming.
Another option is to contact your nearest YMCA and see what they have because they may also have a summer league.
Just found this: http://www.taafregion4.org/Region_4_Swimming.html
Email them at ____@____.com and see if they can help you.
Arlington is region 4, but Grand Prairie is in region 7 and they show a swim team. That may be your best bet if you can't find one in Arlington. http://www.eteamz.com/taafregion7/news/index.cfm?cat=393731