Rule of thumb -- according to the "last action rule", the last action taken by USCIS governs your status. If the last action was approval of H-1B, from the effective date of H-1B, you will be in H-1B status and in your case, you can no longer work for the L-1 employer. Remember, a mere approval of H-1B will not immediately change your status from L-1 to H-1B, the H-1B petition must have been filed as a change of status petition (H-1B approval notice should have an updated I-94 at the bottom) and H-1B effective date must have been reached.
Any employment with L-1 employer after the taking effect of H-1B is concerned unauthorized.
How to overcome that situation? If the L-1 petition is still valid (i.e., not revoked by the employer) and not expired, and if the associated LCA is also valid, you should get a visa on the L-1 prior to entering US and resuming work on L-1.
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