Absolutely a fan of a deep litter situation. Been doing it for years.
First make sure you're not overstocking. Absolute max 1 hen per m2. Second check that they are not in a low spot.
Third, add carbon to their pen!
Initially, I'd put a couple of bails of hay in there (take out the twine) and let the chooks play with them and spread them around. Then as you have it, add grass clippings, leaf litter, saw dust, toilet rolls, ripped up egg cartons (ask your neighbours and friends), dried cow manure from the local farmer's paddock, anything and everything.
I don't use *fresh* wood chip, as I'm convinced (rightly or wrongly) that the hens that scratch well could end up with splinters.
My deep litter pen is my compost bin. Everything goes in there. All kitchen organic waste. This deep litter situation will attract all sorts of life and creepies and the chookies will delight in digging through and their diet will be supplemented beautifully. Typically, I periodically move the pen and its vacated position becomes next season's vegetable garden. The hens enjoy the grass and weeds in the new location until they've almost destroyed it, then I start adding my scraps and compost and we start all over again.
Re sloping the run, I would start with a series of berms, a series of small hills and valleys, especially if the whole area around the pen is level. High points that will dry fast and lower points that will allow the water to drain and you can direct the water from the valleys to your desired exit. Will be easy to achieve with a pick or mattock.
You could create your berms by trench composting in the pen. Dig a trench and drop your scraps in it. Leave it uncovered and the hens will enjoy the scraps. The moisture will drain to the trench too, and beside the trench will dry out. Once there's a fair amount of scraps in that trench, dig another about 50cm from it, parallel to it.
Enjoy the adventure!