This...

My idea for refactoring the class would be to make MyObj abstract, and then creates sub types

...or generic solution...

public class MyObj<T> {
   private T type;
   private String foo;
   private String bar;

   public MyObj(T type, String foo, String bar){
      this.type = type;
      this.foo = foo;
      this.bar = bar;
   }
}

...or without polymorphism, that shouldn't be used for inheritance, have TypeOneBackedMyObj and TypeTwoBackedMyObj with a field, or a property, of type MyObj...

public class MyObj {
   private String foo;
   private String bar;

   public MyObj(String foo, String bar) {
      this.foo = foo;
      this.bar = bar;
   }
}

public class TypeOneBackedMyObj {
   
   private MyObj myObj;

   public TypeOneBackedMyObj(myObj) {
      this.myObj = myObj;
   }
   ...
   ...
}

public class TypeTwoBackedMyObj {

   privat MyObj myObj;

   public TypeOneBackedMyObj(MyObj myObj) {
      this.myObj = myObj;
   }
   ...
   ...
}

...or with polymophism the TypeOneBackedMyObj and TypeTwoBackedMyObj having a field, or a property, of type MyObj implement an interface.

The details strain the solution space.

This...

My idea for refactoring the class would be to make MyObj abstract, and then creates sub types

...or generic solution...

public class MyObj<T> {
   private T type;
   private String foo;
   private String bar;

   public MyObj(T type, String foo, String bar){
      this.type = type;
      this.foo = foo;
      this.bar = bar;
   }
}

...or without polymorphism, that shouldn't be used for inheritance, have TypeOneBackedMyObj and TypeTwoBackedMyObj with a field, or a property, of type MyObj...

public class MyObj {
   private String foo;
   private String bar;

   public MyObj(String foo, String bar) {
      this.foo = foo;
      this.bar = bar;
   }
}

public class TypeOneBackedMyObj {
   
   private MyObj myObj;

   public TypeOneBackedMyObj(myObj) {
      this.myObj = myObj;
   }
   ...
   ...
}

public class TypeTwoBackedMyObj {

   private MyObj myObj;

   public TypeOneBackedMyObj(MyObj myObj) {
      this.myObj = myObj;
   }
   ...
   ...
}

...or with polymophism the TypeOneBackedMyObj and TypeTwoBackedMyObj having a field, or a property, of type MyObj implement an interface.

The details strain the solution space.