Note that if Awesomium is properly distributed, the assemblies and native dependencies should be deployed to the target system's GAC so they don't actually need to be anywhere around the executable.
For details, read: Distributing Awesomium.NET.
However, is you're not deploying to GAC and you need to load the libraries from a location other then the app's root, check this utility.
To use this helper, call
AwesomiumResolverService.Initialize passing the directory with the libraries, anytime before an attempt to load Awesomium (like in your application's