I know I am doing some archaeology reviving a 2yo post, but a detailed answered may be of use for someone else.
So yes, Cefsharp.Offscreen is fit to the task. 
Here under is a class which will handle all the browser activity.
using System;
using System.IO;
using System.Threading;
using CefSharp;
using CefSharp.OffScreen;
namespace [whatever]
{
    public class Browser
    {
        /// <summary>
        /// The browser page
        /// </summary>
        public ChromiumWebBrowser Page { get; private set; }
        /// <summary>
        /// The request context
        /// </summary>
        public RequestContext RequestContext { get; private set; }
        // chromium does not manage timeouts, so we'll implement one
        private ManualResetEvent manualResetEvent = new ManualResetEvent(false);
        public Browser()
        {
            var settings = new CefSettings()
            {
                //By default CefSharp will use an in-memory cache, you need to     specify a Cache Folder to persist data
                CachePath =     Path.Combine(Environment.GetFolderPath(Environment.SpecialFolder.LocalApplicationData), "CefSharp\\Cache"),
            };
            //Autoshutdown when closing
            CefSharpSettings.ShutdownOnExit = true;
            //Perform dependency check to make sure all relevant resources are in our     output directory.
            Cef.Initialize(settings, performDependencyCheck: true, browserProcessHandler: null);
            RequestContext = new RequestContext();
            Page = new ChromiumWebBrowser("", null, RequestContext);
            PageInitialize();
        }
        /// <summary>
        /// Open the given url
        /// </summary>
        /// <param name="url">the url</param>
        /// <returns></returns>
        public void OpenUrl(string url)
        {
            try
            {
                Page.LoadingStateChanged += PageLoadingStateChanged;
                if (Page.IsBrowserInitialized)
                {
                    Page.Load(url);
                    //create a 60 sec timeout 
                    bool isSignalled = manualResetEvent.WaitOne(TimeSpan.FromSeconds(60));
                    manualResetEvent.Reset();
                    //As the request may actually get an answer, we'll force stop when the timeout is passed
                    if (!isSignalled)
                    {
                        Page.Stop();
                    }
                }
            }
            catch (ObjectDisposedException)
            {
                //happens on the manualResetEvent.Reset(); when a cancelation token has disposed the context
            }
            Page.LoadingStateChanged -= PageLoadingStateChanged;
        }
        /// <summary>
        /// Manage the IsLoading parameter
        /// </summary>
        /// <param name="sender"></param>
        /// <param name="e"></param>
        private void PageLoadingStateChanged(object sender, LoadingStateChangedEventArgs e)
        {
            // Check to see if loading is complete - this event is called twice, one when loading starts
            // second time when it's finished
            if (!e.IsLoading)
            {
                manualResetEvent.Set();
            }
        }
        /// <summary>
        /// Wait until page initialization
        /// </summary>
        private void PageInitialize()
        {
            SpinWait.SpinUntil(() => Page.IsBrowserInitialized);
        }
    }
}
Now in my app I just need to do the following: 
public MainWindow()
{
    InitializeComponent();
    _browser = new Browser();
}
private async void GetGoogleSource()
{
    _browser.OpenUrl("http://icanhazip.com/");
    string source = await _browser.Page.GetSourceAsync();
}
And here is the string I get 
"<html><head></head><body><pre style=\"word-wrap: break-word; white-space: pre-wrap;\">NotGonnaGiveYouMyIP:)\n</pre></body></html>"