Given your addiction history it sounds like buspirone (Buspar.) It has little addictive potential and is usually prescribed for Generalized Anxiety (among other things.) It is not a sleep-aid per se, but by ameliorating the anxiety it can be helpful for sleep.
Pull up a picture and see if the pill looks similar. Oblong white pill, can be divided.
BTW, it can be very useful to keep a medication journal. Really helps for situations like this
Hope this helps.