Hi,
except for the overheating problem, I have the same issue with my G73sw :
the laptop confuses
power plug connected with -> says on battery, discharging (no overheating, but the battery is discharging !!
power plug connected without battery -> still works, no overheating, but the battery is not charging
power plug disconnected with battery --> says "power plug connected", but runs on battery, and discharges battery
sometimes after 1 hour the laptop seems to get into "ac connected" mode again, and charges the battery.
I reinstalled several times, but this occurs event with the laptop shut down (the charging light does not get on when plugging the ac adapter).
My battery if fully working when it is charged (10.2% wear only) and the pins on the battery and the pc are clean.
My adapter is genuine, fully working and the pins are clean, the cord is in perfect health.
I tried with another genuine asus adapter too -> same issue.
For me, there is really a recurring a power jack design issue on the motherboard on this series of laptops, that should be seriously be taken into consideration by Asus for the design of the next series of gaming laptops.