Technically, yes, you can stay if the H-1B has been approved, but you would need to 1. have the I-94 that is attached to that approval, as that would be your only means of proving that you are in valid status in the US now once the B-2 I-94 expires and 2. if you are not placed on a project and not working and not getting paid, you are in violation of that H-1B status and will not be eligible for a transfer H-1B. You need to be on payroll once the H-1B becomes effective; otherwise both you and the employer are in violation of the immigration and the department of labor laws.
Please let me know if you have any additional questions or if I may be of further assistance in this matter.
Thanks,
Lana
Immigration Attorney
Admitted to practice in CA, AZ, IN and OH