If you have an approved H-1B petition, you have a job. There is no such thing as "looking for a job". Your employer is responsible for placing you in a position and paying you at all times. If this is not the case with your employer, you have made a poor choice and are likely to have problems in the future.
L1 jobs have prior work requirements. If you meet the requirements with another employer, you may certainly move to that company as your new employer.

