The issues you are seeing aren’t caused by Domain Driven Design, but rather by a lack of separation of concerns. Domain Driven Design isn’t just about placing data and behavior together.
The first thing I would recommend is taking a day or so and reading Domain Driven Design Quickly available as a free download from Info-Q. This will provide an overview of the different types of domain objects: entities, value objects, services, repositories, and factories.
The second thing I would recommend is to go read up on the Single Responsibility Principle.
The third thing I would recommend is that you begin to immerse yourself in Test Driven Development. While learning to design by writing tests first won’t necessarily make you designs great, they tend to guide you toward loosely coupled designs and reveal design issues earlier.
In the example you provided, WebsiteUser definitely has way too many responsibilities. In fact, you may not have a need for WebsiteUser at all as users are generally represented by an ISecurityPrincipal.
It’s a bit hard to suggest exactly how you should approach your design given the lack of business context, but I would first recommend doing some brain-storming by creating some index cards representing each of the major nouns you have in your system (e.g. Customer, Order, Receipt, Product, etc.). Write down candidate class names at the top, what responsibilities you feel are inherent to the class off to the left, and the classes it will collaborate with to the right. If some behavior doesn’t feel like it belongs on any of the objects, it’s probably a good service candidate (i.e. AuthenticationService). Spread the cards out on the table with your colleges and discuss. Don’t make too much of this though, as this is really only intended as a brainstorming design exercise. It can be a little easier to do this at times than using a whiteboard because you can move things around.
Long term, you should really pick up the book Domain Driven Design by Eric Evans. It’s a big read, but well worth your time. I’d also recommend you pick up either
Agile Software Development, Principles, Patterns, and Practices or Agile Principles, Patterns, and Practices in C# depending on your language preference.