Is my AGR career manager the only person who can move me (He is telling me he can't that it's up to my unit)?

HRC and your career manager control your moves. Are you trying to move early or out of cycle? Is it possbile they mean it requires a 4187 requesting to move with you COC recommendation/endorsement on the 4187? If this is your first tour, it is very unusual for HRC to move someone before they have 3 years time on station.

As an AGR myself, my unit can recommend transfer to another assignment. HRC will then see if it's feasible (time on station and all that stuff) and if there is some place to put you. If you have to PCS and your still new, then don't count on it. I find it easier if you are looking to move, find a unit within 50miles (like on a base), and do a 440 order to move you around. But again, the unit has the final word (95% of the time) to release you. But that's just my observation.

It's up to HRC and your SGM. Speak to your Chain of Command for assistance.