If I re-enlist and get orders to go somewhere else, would I still have to do my remaining time at my current duty station?

When you request a duty station through reenlistment you will receive an assignment with a report date that meets the time on station rules. In your case, it will generally be 2 years on station.

Based on your profile saying you're in the 101, if you have 24 months time on station (usually 12 months is the minimum), they will PCS as soon as they have a need at the next duty station. I say a need because for overseas locations it's based on DEROS (date expected return from overseas) because it's a one for one basically. I would imagine that this within 3 months.

If you re-enlisted and then came down on orders...you move out based on the dates on the orders. You would stay on your current station long enough until it's time to clear then move out.