Sounds like a potential issue with a "door ajar" switch. These are switches mounted on the door latches. When the door is not latched entirely or at all, the switch tells the dash and overhead light to stay on.
Even with the doors shut, a switch may send a faulty signal, sticking in the "door open" position. To try and un-stick it, lube the latches on both doors very well, then open and close them a few times. Often times this works to turn the dome light out, etc.