Retail Girl, this is the most straight forward advice I can give you in regards to the LOD vs. GSA debate. Follow the orders of the highest ranking person, if not then you could be terminated. A GSA is the same level as you. The LOD is in charge of the store at the time and you should follow their instructions (of course use common sense and weigh the words of the STL in comparison to the flow TL for example). A GSA may think they know best but if they are trying to tell you to ignore the LOD then it shows they know nothing instead. When I am tasking out a TM or checking to see what they are working on and they inform me someone else has given them direction (be it another TL or the STL) then I might change the directions given. If I do I explain to them that I will go talk to the person and let them know why the instructions are being changed and that I will take full responsibility and the TM will not get into trouble, I also explain that we will still get the project completed and explain why the other project is taking precedence. Most of the time I won't change the directions because of a respect thing, unless it's just busy work or they are goofing off, etc.
For everyone about to spout off crap about how it's done at your store, you might as well save it. Don't get her in trouble because of your broken stores.