Scott,
The Lower Illinois is the closest to Tulsa. The problem with fishing it right now is generation from Tenkiller Dam. Here is a site that shows the flow rate for the river http://waterdata.usgs.gov/usa/nwis/uv?site_no=07198000
Normal flow is around 100 CFS and it is close to 4000 CFS right now. Here is a link to the Dept. of Wildlife Conservation http://www.wildlifedepartment.com/ It has information about licenses. Another option would be Pawhuska Lake northwest of Tulsa. It is a winter only trout fishery and will have trout at least until April.
Steve