For singles play you can try tennisleague.com. You don’t need to be a USTA member for this, but leagues are organized based on USTA NTRP ratings (and the number of games one plays in a season just depends on how many matches one can fit in). So there is no requirement to play every week/weekend. Most players tend to play at their level in the singles leagues, unlike USTA tournaments where there are always players playing a level lower to go deeper into the tournament.
You can find information about USTA leagues and tournaments here:
http://www.usta.com/TennisLink/Default.aspx
You can search for USTA tournaments here:
http://tennislink.usta.com/tournaments/Schedule/Search.aspx
another league to consider is world team tennis their website is:
http://www.wtt.com/
For singles play there are other leagues too like Ultimate Tennis and the USTA’s Flex League.
