2 flashing red lights, but in a different pattern than "Overheating"

Last night our Xbox 360 started displaying 2 flashing red lights. It was very odd because the console had been off and no one was playing a game, but suddenly it turned itself on and the 2 lights started flashing. We tried to shut it off but it wouldn't power down. I connected a controller to see if everything still appeared to work, and it did... no error messages, no weird behavior. A quick Google search suggested that the 2 red lights were indicative of an "Overheating" error, so we unplugged the console and set it aside overnight.

Today I plugged the console back in. IMMEDIATELY upon connecting the power cable, the console turned itself on and flashed 2 red lights again. It seems HIGHLY unlikely to me that we have an overheating issue when the console had sat, unplugged and disconnected, for about 16 hours, yet the red lights activated the second the power cord was reattached.

After more internet searching, it seems like the overheating issue is associated specifically with the lights in quadrants 1 and 3 being red (top left and lower left). This is not the case with our console. The lights that flash red have been in quadrants 2 and 3 (top right and lower left) or quadrants 1 and 4 (top left and lower right). I have been unable to find any information about what this might mean. The console still appears to work and we still haven't seen any error messages. However, the ONLY way to get the console to power down and stay powered down is to disconnect the power cord. If I turn it off, it turns itself back on within 5 seconds (usually almost immediately). Also, I tried the process for determining the secondary error code by holding the sync button and pressing the eject button. This did nothing - no flashing pattern, no other information... all it did was make the disc tray pop open. Therefore, I have no 4-digit secondary code to give as additional information.

I should probably mention that one of our dogs seems to have peed all over the console... *sigh*. Any chance this is some weird error code that means something inside is damp? Perhaps if I let it sit, disconnected, for a few days it will fully dry out and be ok?

Any help or information will be appreciated. Thank you.