The GC and the H1b are pretty much independent. Nothing has to be transferred, you might want to get copies of all approval notices and all filings sent out by your current shyster in case there are questions.
You just go to a new lawyer, you file a form G28 with your GC petition and for this case, the new lawyer will be your representative (if there are questions regarding your H1b, the service will still contact the old attorney as they have his G28 in the paper file and seemingly no computerized system to update all applications with tne new G28).
