Shavings are terrible for composting, in that they take forever to break down. The nitrogen is likely fixed by the chicken manure, but it adds a level of doubt that doesn't really need to be there. I use to use them when I first started with chickens because I didn't know any better and that's what was available at the feed store. I currently use a mix of wood chips (not shavings but stuff that is sent through a chipper), grass clippings, dried pine needles, dried leaves, a small amount of chopped straw (that I use in the nesting boxes, but it gets mixed into the bedding by the birds on occasion). I use the same basic mix in my coop and my run - my birds aren't 'locked' into their coop at night, but rather run and coop are secure - you can see my setup by following the link in my signature.
When I do maintenance, I move the bedding material from the interior coop to the chicken run for it to finish composting and replenish the bedding in the interior coop with a fresh mix - I start with a layer of mixed sized wood chips then add the rest in as it becomes available. The birds do an amazing job of breaking down the wood chip based mix, especially in the run. I add organic material to the run regularly to keep the birds busy doing my compost work. I also compost, for the most part, in my run. All of my garden waste goes in the run, as well as a variety of table scraps. Anything I don't feel is safe for the birds goes into a traditional compost bin outside of the coop. I also occasionally compost expired birds in my exterior compost bin - anything that dies of disease or questionable causes goes off to the state lab for necropsy and doesn't make it to the compost bin.
Many 'Tree Service' businesses will deliver piles of wood chips for free if you have room to have it. It's easy to get a perpetual supply for basically nothing.