I hope you received the reply to your first question, I sent it last
night.
I would put the quarters in your refrigerator in pans/tubs without
bagging until they are chilled through, (6- 8 hours ?) Then I would place
them in clear plastic bags to prevent excessive drying out of the meat,
until you finish cutting them up.
I compare skinning a deer to peeling an apple or a banana, once the
outside protection (skin) is remove the dehydration process is accelerated.
The longer it sits the more it dries out, as it the meat of the deer dries
out it starts to toughen. Your options are to process it soon before it
toughens, or leave it for 10-14 days so that the enzymes and bacteria action
breaks down the cells (aging meat) and makes it tender again.
I prefer to process it soon (two days is fine) as my preference is
"fresh" meat over aged meat. There are many arguments both ways, and it's
strictly up to the individual as to their preference.

Dave