First, isolate the cause. Fully charge each battery, then test the camera without the battery grip using one battery at a time. Check the camera’s Battery Info screen to note the charge level, leave the camera off, and see whether the battery loses charge only in the camera or also when sitting out of it.

If a battery discharges on its own, that points to a bad battery. If batteries only drain while installed, the issue is more likely the grip, camera body, or a short/power drain somewhere.

Also be careful not to remove/open the card door immediately after card activity. Some users report that the 5D Mark III may still be finishing card access briefly after the activity light stops, and interrupting that can cause the camera to lock up and drain the battery.

Because you’re using LP-E6 batteries and the LC-E6 charger, check whether your batteries/charger are affected by Canon’s advisory for those models.

In short: test batteries individually, test without the grip, compare charge loss in and out of the camera, and verify whether your LP-E6/LC-E6 equipment is covered by Canon’s advisory.