We are you moving to? Raleigh, Durham? I work for IBM and we have our own sports leagues so you can always go that route also...
I know Raleigh has a very competitive softball league... There are several tiers... A, B C... all the way down to G I believe.
Just did a quick google search and came up with these. Good luck and welcome to the area.
City of Raleigh | 2007 Adult Registration Dates (broken link)
Parks and Recreation : Athletics : Spring Adult Open Basketball
If you are looking to meet new people in the triangle I would highly recommend Tri-Sports... Not sure how only you are, but this is a group of 20 or 30 somethings looking to have a good time, play sports, hang out, etc.
TRI SPORTS, where Triangle professionals come to Meet and Compete!
I hope this helps!