From the facts you have given, I do not see why you have to go through the quota again. You can do an H1B transfer as long as you can show an approved I-140 and that you maintained status under the adjustment of status (EAD as proof). However, there might be other issues involved in your case that the attorney is saying regarding getting back on your H1B extension. Talk to a good immigration lawyer.
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The new employer can petition for H-1B status, but you will not be approved as a change of status. You cannot change from and EAD to H-1B. Usually, CIS will only approved a change from one nonimmigrant (temporary) status to another. CIS should approved the H-1B petition, however in order to "activate" this H-1B status you would need to leave the US, get a H-1B visa at a consulate, and return in H-1B status.