When the 6d came out Canon marketed it as weather sealed.
"Dust and drip proofing protection around control buttons and along outer cover seams has been upgraded from the EOS 5D Mark II and is equivalent to the protection on the EOS 5D Mark III."
I would contact Canon and hear what they have to say. Given that the rear screen is fogged. They might be fussy if you used it with an un-sealed lens, which is unfortunately most of their lenses since as I recall Canon only seals their L lenses (and some of those are not sealed either). Pentax and Nikon tend to give much better sealing across their non-pro lenses.
With regards to the error message....
Description: A malfunction with the mechanical mechanism has been detected.
Resolution: Turn the power off, then remove and re-install the battery. Then turn the power on again
From their knowledge base:
"Err 20: Mechanical malfunction Within the mirror, shutter or aperture mechanism (most likely), some sort of disturbance, error or lock-up has been detected. This error code doesn’t specify where the exact problem is, but points to the cause being mechanical rather than an electronic gremlin."
From this I would guess you did more than go out in the rain. If I were a camera tech faced with two errors on the same camera, I would guess abuse. In this case water immersion. If this is not true, then you can look inside the camera housing for clues. Does the mirror go up and down, etc. I would also rice it again and store in a warm dry place (dehumidifier).
If all that fails, send it in to a repair shop for an estimate. (and this is where you are almost certainly going to end up). I suspect the camera is going to be scrap. If you want a low cost wet-weather camera (for less money) get a Pentax. If you want a system more in line with Canon but more sealed choices (and better low light performance), go Nikon.
Not what you wanted to hear, I suspect, but I hope you find it helpful at least.