I don’t know how air-tight that shed is. In any case it will help, I just don’t know how much “draft” you might get through there. Also, chickens create a lot of dust. Whatever is in that shed will likely get a coating of dust. My coop is the closed off end of a shed and I have openings into the shed, but the shed is not anywhere close to airtight. I have a large opening up high and another down low. The down low is really good in the summer because it is in the shade so the air is cooler. And I’m OK with the dust in there. Those openings provide a lot of ventilation without creating a breeze.
I don’t know how far along you are on the build, but I strongly suggest you put a sloped roof on your coop. A flat roof will pond rainwater. It will either rot the roof or leak through, if not both. You need to direct that rainwater runoff away from your run to help keep it dry and do not put a door where rainwater will run off onto you when you are going in or out. Gutters and a downspout may help.
The idea is not to keep the coop warm, it’s to allow your chickens to keep themselves warm. They wear a down coat so they can do that. And don’t think of a draft as a tiny leak around your house windows that robs you of energy to heat and cool your house, that’s not a problem. What you want to avoid is a breeze strong enough to ruffle their feathers which allows heat to escape.
If you put a decent slope on that roof with a little overhang at the bottom and more overhang at the top to keep out rain, you can have an opening above both walls under that overhang and put the roosts under the high end. Any breeze should be over their heads.
A small coop like that can be a big challenge. At least you don’t have a 4x4, those are even more of a challenge but some people make even those work. While some people like the nests up off of the floor (I do too) you may not have that luxury. As long as you have a high enough lip to keep the bedding and poop from being raked into the nest when they scratch you can put them really close to the floor to save vertical room.