Bernd,
I'm not sure if I correctly understand what you mean, so maybe this is beside the point.
When you have a class with various attributes, all of which can have a finite number of values, than a state machine for that class would have a region for each of these attributes, and in each region it would have a state for each value that the corresponding attribute can have.
Now for an instance of your class you can only set one object state, and anyway you cannot create regions directly in a state machine diagram, because EA does not support it. The neatest thing IMO would be if EA let you associate a state machine (not a diagram, but a MACHINE element) to your class, give it regions, associate the attributes with the regions, and in an instance let you set an object state for each region.
But this is just a dream.
In reality all I can think of is this (David already mentioned the general idea): Create enum classes, give them each a state machine diagram with states representing the values they can have, and give your class an association to each enum class. In the object diagram (or wherever you want to show your instances) create an instance of the class and an instance of each enum class, associate, and set each enum class instance the desired object state from its state machine diagram.
Frank