Can I transfer H1 immedietly after approval,not joining the orig. Sponsor

I am in L1, aplied H1 COS thro consultancy, its in process. in the mean time I got offer from another employer who is ready to transfer my H1 once its approved. Now 1). Should I Join the consultancy before applying transfer? 2). Can I continue to work on L1 and wait for H1 transfer to complete and directly join the Employer instead of joining consultancy and transfer?