Oil cooler failure

Unfortunately the oil cooler is part of the radiator so it could be the issue.

Here are a couple guides that will help with this:

https://www.2carpros.com/articles/coolantantifreeze-in-the-engine-oil

https://www.2carpros.com/articles/head-gasket-blown-test

Once we know the results of this we can better determine is there is another issue.

However, if the gasket is okay then the cooler is the issue but you can pressure test the cooling system to confirm this.

https://www.2carpros.com/articles/radiator-pressure-test

Below is the process to replace the cooler but it is part of the radiator so you need to replace the radiator.

Let us know if you have other questions. Thanks