id probably stick to the rock river...i've never fished it but have heard good things....i would attack it as follows....head out mid afternoon so you have enough time to scout and potentially catch bait if thats legal (i always bring some with since i have really bad luck when i'm depending on trying to catch bait on the river for the night). i'd be looking for deep holes, holes and undercuts on the outsides of corners, and wood piles. the bigger the brush piles and the deeper the hole the better, preferably wood and deep water in the same area. i would be marking way points on the gps during the day so i know which ones i'd like to try, get familiar with the river, and make sure you mark or remember where there are hazards in the river. i've been out plenty of times when the fog rolls in on the river and you can't see more then 10 feet in front of you.
i would spend up to 30 minutes per hole, if there are no bites, move along to the next spot. if you're strictly looking for flatheads, i'd stick with live bait (baitfish, bluegills, bullheads, small carp or sheephead, whatever is legal to use down in illinois). or if you want to pass the time and you have some dead bait, cut it up and use that....you'll more than likely find some channel cats to pass the time and every now and then a flathead will take it.
i'd be anchoring 10 to 20 yards up river from the cover and/or hole and cast down to it. up in wisconsin we are permitted three baits in the water, so early in the night i put one bait close to shore, one midway out, and a third out toward the tip of the wood. later in the night i would be putting two near the wood and a third on a shallow flat if there is one near by....later in the night the flatties will sometimes come up and cruise the flats looking for food.