Will I be able to submit a greencard petition and work legally in the US if I start with a new employer? I’m in the sixth year of my H1B.

Your only hope of being able to remain in the U.S. and continuing in H-1B status is if your Immigrant Petition is approved prior to the expiration of your H-1B visa (post reclamation of time). It may be too late to rely of the rule that allows for extensions based on a pending PERM application since you are already in your sixth year. There are other options in such situations including switching to a different status (F-student) and then returning to H-1B upon I-140 approval. However, it is best if you discuss this with an Attorney based on your specific circumstances.