Hmmmm, there is this -
Auto Move Closed Threads which gives a new option to close and move the thread at the same time. You can see how that is done and then just change it to close and unstick.
Do your mods make a final post in the thread as they close and unstick it? You may be interested in one of these mods if that is the case -
Close/Open/Sticky/Unsticky via Quick replay or
cLosed , Open , Sticky , Unstick - On Quick Reply - Edit Template