The camera's battery died yesterday. I changed the batteries this morning and now I only get the green light to come on, but the light stays on when the camera is off as well

It sounds like your power on button may be stuck, if the ON indicator is always on. That may be the reason your battery was dead yesterday. Try reinstalling your batteries, checking carefully the positive and negative positions, and working your power button several times rapidly to see if it is stuck. If it is still showing a constant ON condition, try the following steps. With the batteries out of the camera, try a tiny bit of rubbing alcohol around the power switch, working the switch rapidly for a few seconds to see if that frees it. Replace the batteries with known fresh ones and try to power up the camera. That may fix your problem. If not, the camera will probably have to be repaired or replaced.