I would stay with consultacy if they are good. You can ask them to start GC immediately. I did the same when I was offered fultime. But, if your contract will be terminated if you dont join FT, you may not be able to find another longyerm project immediately which may affect your GC processing as your consultancy might say that you dont have project. Also, you need to file H1B again. Regarding resume, you can give the updated resume deleting fake things if you decided to join.
This is just my personal opinion