It sounds like your apartment complex has been "master-keyed". That means every lock in the entire place typically can be opened with the master key (the key your property manager has) and a control key (the key you get). The master key opens any lock where your control key only opens your lock.
Master keying works because there are extra pins put into your lock's cylinder to allow multiple combinations of key shapes to line up all of the pins.
If the ORIGINAL locksmith that master-keyed the complex rekeys your lock, it is likely that the master key will still open it. If you call another locksmith that is unfamiliar with the master key chart, they will likely just throw away all pins and start over. In that case, the master key will no longer work in your lock. Rekeys typically cost about $70-$80 for the first lock & $20-$30 for each additional. Trip fees are usually built into the first lock's service.
New locks will solve your problem also and might even be cheaper, but then you have to do all of the work yourself, and it is not likely you will be able to find locks that match the look of the other locks in the complex. It sounds like your property manager is okay with you rekeying the locks (with his little disclaimer) but they're usually not okay with installing locks that don't visually match the originals.
Hope that helps.
Pop-A-Lock of San Francisco