Hi! I have several GSTs, and mine sag slightly as well. Mine sags mostly in the back of the bag, and I think I finally figured out why! It's because I put a lot of "heavy" stuff in the inside back zipper pocket such as my keys and 3 lipsticks. Stick you fist down in that back inside zipper and you can see what I mean). The front of my GST looks fine, no sag. I think it's because i never use that inside front little pocket.
That said, I do not use a purse insert as I think they look kind of cheap (my opinion). It also takes up space, especially for a GST since it is already divided. I sent two GSTs to have then cleaned and also asked they fix the sag which they did. It looks great and is super straight... almost a bit too straight, but it looks crispy.
Now that I have two GSTs that stand tall and several others than have a slight sag, I actually do not mind a slight sag. I think it gives the bag that "cool used luxury bag vibe". So, as long as my GSTs don't sag too much, I will leave it as is. If they start to get really bad, I will send it in for a cleaning and stiffening.
I think if it really bothers you, send it in to get it fixed. I think bag inserts will take away from the limited space of the outside sections. And if you put in two inserts, you will now squeeze the center zipper section and take away from that open space.
I hope that helps with your question in deciding what to do.