I definitely do not have the level of experience that you have so take this with a grain of salt. I have seen and worked in a lot of companies that have a very heavy portion of their code base in web forms. Many of these companies have been working to transition to MVC but it takes a longer time the larger your legacy beast is.
I would suggest finding a job where you can leverage your experience in .NET Web Forms, while picking up the pedigree for MVC.
I would suggest finding a job where you can leverage your experience in .NET Web Forms, while picking up the pedigree for MVC.