Consider St. Louis, especially if you're a baseball fan. Unlike the Chicago teams, the Cardinals win a World Series now and then. Coordinate your trip with a baseball home stand that has tickets available, and you could golf during the day and watch baseball at night.
St. Louis area has lots of good to excellent-quality public and semi-private courses on both Missouri and Illinois side. Many are run by Walters Golf Management. http://www.wgmgolf.com/?page=35211
We have some signature courses in the area: Stonewolf GC in O'Fallon, IL by Nicklaus, and Spencer T. Olin GC in Alton, IL by Palmer. Hale Irwin did the layouts at Forest Park and Quail Creek GC on St. Louis side. Missouri Bluffs GC is a frequent site of PGA and USGA qualifying tournaments.
Normandie GC, designed by Robert Foulis (he apprenticed with Old Tom Morris at St. Andrew's), is the oldest public course west of the Mississippi.
Green's fees (cart included) would max out at about $75 on weekends, with lots under $50 (and cheaper) on weekdays. Also, you can get good-quality hotel and motel rooms for under $100 (if there's not a major convention in town).