Target small companies. Very small. Companies which are unable to hire two people in their IT department. Explain that you're skilled in both development and design, and show them some proof.
The larger is the company, the more specialized is the staff. When the budget for a project is high, a company can afford hiring a visual designer, an UX expert, several developers, each of those developers being specialized in a specific domain, DBAs, a few consultants, managers, testers, etc.
Small companies just cannot hire lots of people. They want to hire one or two who will have to do all the work. On the other hand, they do not require the person to be an expert in a domain. They look for a jack of all trades, master of none, i.e. people who don't have a deep experience in a specific domain, but know lots of stuff an expert will never know about things which are not in his domain of expertise.
The situation with developers vs. designers is slightly different. For a person with no technical background, a tester still works with code. A DBA writes code. A person who types HTML types code. They are all coders, i.e. the same as developers. Designers are a different kind of people. They don't type. They work with Photoshop.
Technically, it's wrong. An expert DBA has nothing to do with an expert C# developer neither. A tester doesn't work much with code, and especially doesn't write code. But for ordinary people, only designers are apart, since they understand this distinction, but don't get any other (and mostly never heard about UX experts for example).
It means that they will find it strange that a person does both design and development at the same time. They will have doubts that you're competent in either. But if you explain that you're passionate about both, show that you have enough experience, etc., it may be tempting for them to hire you, instead of hiring two separate people to do the same work.
Also, chances are, an existent small company has already a developer and a designer. If they are looking for a new developer, they still probably want to keep their designer. If they fired their designer, they still have their developer. This means that you'll have better chances to find a job in startups, where they don't have anybody for the moment.
But wait! Are you sure you want to work for those companies? In general, it's not very tempting to work in companies where there is no enough money to hire several people in IT department. It may be a temporary job, but still... Forget about Joel test. Forget about working with more experienced colleagues and learning from them daily. Forget about writing high quality code with pair programming, code reviews, etc. The company may even refuse to invest in a bug tracking system or even a source control.
Instead, what you can do is to:
Either work as a freelancer (if it's possible for you), and to suggest your services both as a developer and as a designer,
Or find a job where you're more specialized, but where competencies from the other sector are welcome. For example, searching for a visual designer is not the same as searching for a web designer who must know how visual design is implemented, how it will affect the website performance, how it must be optimized, etc.