The approach with injecting different connection string to access different databases with one and the same schema is valid and it should work.
The exception you mention indicates that you are trying to use a class in the GetAll() method which most probably is defined in another OpenAccessContext. To work in this way you should have a single model and mapping for it.
How many mappings do you have in your project? What kind of mapping do you use (RLNQ, Fluent or Attribute)?
OpenAccess ORM is now Telerik Data Access
. For more information on the new names, please, check out the Telerik Product Map