First, a graphic designer in Qingdao, in order to make any kind of decent money, would have to have a sufficient command of the Chinese language. I would be very hard for a graphic designer in that city to live off the expat community alone.

That being said, if I were looking for a graphic designer position in Qingdao, I probably would start by putting up my resume on this site here, and on thebeijinger, and on shanghaiexpat, and other sites like that.  That a least might give you some idea.

Secondly, in terms of other job, there are a great deal of teaching jobs in Qingdao and those are readily available by using this board, and checking the major teaching portals in China.  In this regards, www.anesl.com comes to mind.

Thirdly, believe it or not, there are all kinds of shipbuilding jobs available in Qingdao for qualified tradesman in the trade and there is  a small but steady expat community of shipbuilders in Qingdao -- Brits, Scandinavians and Finns for the most part. For that however, someone needs to be a truly skilled expert in the field.