I'm sorry, but your coop is WAY TOO SMALL no matter how much outside run you give them. It's not 5x6, it's 4x6 -- because the nest boxes don't count. 4x6 is big enough for 6 chickens, not 12 and even for 6 chickens that coop (like all of those "Amish" style coops), has inadequate ventilation.
When considering space you can think of a chicken as a 15" cube -- about 12" of actual chicken and a bit of personal space.
The Usual Guidelines
For each adult, standard-sized hen you need:
- 4 square feet in the coop,
- 10 square feet in the run,
- 1 linear foot of roost,
- 1/4 of a nest box,
- And 1 square foot of permanent, 24/7/365 ventilation, preferably located over the birds' heads when they're sitting on the roost.
For 12 chickens you need:
- 48 square feet in the coop. 6'x8' is more practical than 4'x12' since a long, skinny coop like that would be difficult to work inside.
- 12 feet of roost.
- 120 square feet in the run.
- 3 nest boxes.
- 12 square feet of ventilation.
You will need ventilation just as much in the winter as in the summer. This is BYC's best article on chickens in cold weather:
https://www.backyardchickens.com/articles/cold-weather-poultry-housing-and-care.72010/
One thing you could do is to convert the little run into a coop so that you could divide the flock. You can see what I did to make an outdoor brooder here:
https://www.backyardchickens.com/articles/run-to-outdoor-brooder-conversion.76634/
As for the odor, with the vegetation essentially destroyed you need litter to absorb the poop and compost with it. Wood chips, wood shavings, straw, pine straw, dry leaves, or any other form of dry, organic material. The coarse wood chips you can get by the truckload from a tree-trimming service are often considered the gold standard as run litter.
That's only 2 square feet per chicken, twice the density recommended. Better than commercial caged layers, which get only 1/2 square foot per bird, but far from optimal.
The tighter the chickens are kept the more sanitation and behavior problems are likely to occur.