You can certainly do a change of status from L2 to H1 if you meet the requirements. If you go this route, the expiration of your L2 will not impact the time you have remaining on the H1. As long as your L2 visa is current you are able to travel freely although any lengthy period of out of the country (think 6+ months) may impact the admission period of your new visa.
To know the best plan of action would be to contact an immigration attorney. You definitely want to be sure that you don’t have any missteps that set you back in the process. Plus, a lawyer can guide you in the best direction based on specifics of your situation.
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