I will list for you the trees/shrubs that are thriving in our heavy red clay and maybe you'll get some ideas from the list. Our rainfall is about the same as yours--I'd say 36" is roughly our average, although we vary widely from year to year. In the last decade, our lowest rainfall has been 18" and our highest 51" (this year).

Pretty much anything you plant might need irrigation during dry spells the first three years, but after that they ought to be able to handle almost anything except the most severe drought conditions.

Red Mountain Cedar (Juniperus virginiana)--Because it is so invasive, I hate to recommend this one but it is a tough son of a gun that can handle almost anything the weather throws at it in southcentral OK. I am a little concerned that a severe ice storm might bring down branches from this tree, but ist hasn't happened here (we are so far south that we have escaped the damaging ice storms that have hit OK in the 2000s). Maybe someone who lives closer to you geographically can tell you if the recent ice storms have destroyed cedar trees.

I also wouldn't plant cedar within 100' of a house or other building if you're in an area prone to wildfires/grassfires because their wood can explode and send burning sap through the air and it can land on buildings and ignite them. We took out all the cedars within 100' of our buildings for this reason

Still, cedar is the toughest tree around and if left unpruned the limbs go all the way down to the ground so it is a great windblock/privacy screen. Most have small blue berries. Evergreen. Will reach a mature height of 30-40' and a spread of up to 20', but it will take them a while to max out. They can handle full sun, part shade and dappled shade. This probably is the most widely adapted tree in most of our state.

Hollies in general would work, and really large ones that get almost tree size would block a lot of wind. Nellie R. Stevens is a great one, so is Burford Holly and there are many other hollies. They grow well here in clay that's only been slightly improved with a little compost. Most have berries, but with some varieties you have to have male and female plants to get berries. Many are Evergreen. Here in southern OK (and I am assuming in other parts of OK as well) we have native Possumhaw hollies (Ilex decidua) that are deciduous if you aren't looking strictly for evergreens for a year-round privacy scree. There are some improved cultivars of Possumhaw Holly available commercially. Yaupon Holly (Ilex vomitoria) also handles clay very well and can handle clay soil, and grows in anything from dappled shade to full sun.

Both American Elm (Ulmus americanus) and Cedar Elm (Ulmus crassifolia) are native here in our red clay soils. They provide lovely fall color and can grow in full sun to part shade.

Our woods are full of several kinds of oak that are well-adapted to heavy clay soils and to our hot summers, including post oaks and red oaks. We also have a lot of native bur oaks (Quercus macrocarpa) and they are gorgeous with their huge leaves and acorns. They get really, really, really large though so if I were using them on a property the size of yours, I'd take care to plant them pretty far from the house and other buildings and well away from power lines. It will take them many years, but they'll eventually attain a height in excess of 60 to 80 feet, so you don't want to put them too close to a building because eventually they'll loom large over those buildings and dominate the landscape visually. Our 8-year-old burr oaks that we transplanted from the woodland to the yard when they were about a foot tall are now about 18' tall, so they are not tremendously fast-growing.

Another deciduous tree that grows wild here in heavy clay soil and which survives on nothing but rainfall is the Persimmon. These grow pretty fast and have lovely golden-orange color in the fall. They produce smallish persimmons, which seem to be one of the coyote's favorite (or, at least, most readily available) foods.

Rusty Blackhaw Viburnum (Viburnum rufidulum) is a lovely native small tree/large shrub. It is deciduous, has lovely creamy white and aromatic flowers in spring, and attractive berries in fall. The fall color is pretty good most years. All the ones on our property are understory trees in our mixed oak-hickory woodlands, so they tend to only get 12-15' tall or maybe 18' tall if they have some full sun part of the day. In more open areas, they can get up to 30' tall eventually.

Chaste Tree (Vitex agnus-castus) is a great shrubby tree that has lovely violet-blue flowers in the summer. It will spread out about as wide as it is tall although some people prune them up more into a tree form. There is a pink-flowered form but I've never seen it here.

Native plums like Chickasaw plum or Mexican plum (Prunus mexicanus) are very tough and can handle clay quite well. Their blooms are lovely and scent the air in early spring. Some years you even get small plums from them that you can use to make wild plum jam, but often they bloom too early and a late freeze takes out the flowers/fruit. They tend to get about 15' tall in general, but there are a few very mature ones in our part of the county that are 30' to 35' tall.
Any poplar you plant will be very prone to breaking when there are severe windstorms or ice storms. So would willows. They grow fast, but that also means they decline and break quickly too.

My personal opinion is that planting bamboo would be a big mistake. Bamboo is highly, highly, highly invasive. If you plant it near a property line and it creeps into the neighbor's yard, they can hold you financially responsible for the costs incurred in removing it, and it is difficult to remove.

There are many ornamental grasses that could serve as a mid-height privacy screen interplanted with or adjacent to taller deciduous trees and most ornamental grasses tolerate clay very well (unless you choose one that is native to sandy soil).

The Oklahoma Forestry Dept. sells trees to property owners in bundles in the late winter or early spring for conservation use, windbreaks, wildlife forage/habitat, etc. I'll see if I can find and link that portion of their website.

Southern Wax Myrtle (Myrica cerifera) is another possibility but I do think it would be more likely to suffer ice storm damage as far north as you are. Even when we have only a couple of inches of snow here, I run outside with the rake and knock the ice and snow off the limbs because they start bending under the weight of the snow and I don't want them to break. They are growing just fine here in unimproved clay. Evergreen and has gorgeous red and cream-colored berries. Dwarf Wax Myrtle (Myrtle pusilla) might be a great understory planting to accompany taller trees. They top out at about six feet in height but otherwise are very similar to standard Wax Myrtle.
We have many other native trees and shrubs in our woodland that thrive in clay soil and are ableto survive only on natural rainfall, including honey locust, black locust, native pecan, Texas ash, blackjack oak, chokecherry, Oklahoma redbud and American beautyberry. Some of those probably will be on the forestry list when I find it and link it and you might research them to see if any of them appeal to you.

Dawn

Here is a link that might be useful: OK Forestry Dept. Tree Seedlings

As usual, Dawn has done such a thorough job that I can think of only a few things to add. If you aren't too tied to the evergreen idea, you may find some deciduous shrubs that would give you spring flowers and summer privacy when the leaves are on. I like forsythia, spirea, kerria, lilac, mock orange, and crepe myrtle. Those interspersed with the hollies would give a varied hedge. Here in eastern Ok all these shrubs get to be at least 7 ft tall, and some 9. The native spicebush is also lovely, with golden fall color. And elderberry has lovely spring blooms followed by berries that the birds love. One of the large tough roses would cover the fence. I have New Dawn and Betty Prior, very disease resistant, both.