Sorry, forget last one. Take a look at this.

Basically, you can't and you can not even move to H4 because the 6 years limit is for H category visa totally, including H1B and H4. Without engaging into GC application and reaching certain point, the only option to get another 6 years is to leave the country for a year.

Sorry to tell you because this is happening to my cases too. If you still have at least a year left on your H1, try to start your Green Card application.