We can figure this out. If the coil stays energized the battery is powering the relay. Check as mentioned already each of the 2 screws holding the battery bracket has a shouldered fiber washer, and there is a insulating piece of fishpaper under the battery bracket. After that is confirmed, look closely at the lower solder lug that the ground wire is connected to and it is not touching the battery bracket. Check the bare ground wire is not touching the second solder lug on the bottom of the solder lug stack. The ground wire needs to be the first wire on the stack, looking at it head on. Is the screw holding the solder lug / insulator stack the original one? If the screw has been replaced, check to see if it is too long and punctured the paper insulation wrapped around the coil, the screw will ground the coil to the frame. Sloppy soldering on the terminals on the solder lugs. check to see if any solder has bridged two lugs together. Also check carefully that two lugs are not bent and touching another, (look closely at the bottom lugs). Check that the 2 wires coming out of the relay coil are insulated and not touching the battery bracket.
It's got to be one of those items. Good luck!