Other than the Bartram and Blackwater?
The Perdido (al/fl state line), Escatawpa (al/ms state line), Juniper/Sweetwater - a good one for smaller yaks/canoes (fl), Coldwater (fl), are all solid choices for kayaking/canoeing. They are all flat water but can be swift moving at times. Not sure where you live but those are all within an hour or so to Mobile.
Further east into Florida or Lower Alabama there are plenty of rivers. The Chipola south of Marianna is an awesome river, one section has a nice class 1 drop and there are lots of runs of minimal whitewater. The Holmes and Ecofina creeks north of panama city are spring fed and are really scenic. Supposedly there is a river around Brewton that runs through a quarry that has a little whitewater, i think it is either juniper of burnt corn creek.
if you are looking for specifics i would recommend picking up a few books,
http://www.amazon.com/Pad...a-Joe-Cuhaj/dp/0762721928 and
http://www.amazon.com/Can...qid=1276872983&sr=1-6some websites.......
http://www.dep.state.fl.u...lewest/panhandle_west.htmhttp://www.paddling.net/places/AL/