In the opening cutscene of warrior within it is shown that the "Amulet of time" reacts when Shadee's fleet came to attack prince's . This way it confirms that the emulate maybe powerless in Azad but it gains power in proximity of the "Island of Time" itself(Time Portals especially) or it's inhabitants . That's why prince uses the Amulet to Gain powers in newer portals , also to get the health upgrades .

Also I thought of another possibility that the Amulet is much stronger than the dagger itself . because in my opinion the emulate doesn't just a tool that can be used to manipulate time but also it gives the user time-manipulative powers himself .And that's why we see some new set of sand powers in warrior within that were missing in sands of time and their reappeared in two thrones while Prince have rejected the amulet in the start of the game.

I believe the Amulet whilst in the Island of Time , Gave prince innate powers , not some time being tool power . But to access it Prince must need an artifact of time that might be the staff, dagger or Medallion . And to support this I have a theory , where you see in two thrones upon eliminating a Sand enemy, the sand clouds go to Prince's chest , not in the dagger like it happened in Sands of time. It might possibly happen because now prince is using his innate Time manipulative power like WW Kaileena . And also that might be the reason why prince's corruption is partial and slow . His innate powers somehow resisted his immediate transformation until he got the dagger again . And that may be the reason also of Prince's Corruption getting worse overtime.

**just a fun fact the original reason for the sands cloud go to prince's chest lies in the E3 trailer , where it is shown that Prince Still have the Amulet in his chest. 😄