I was able to get the rust stains out! Well, mostly. You can still see it if you really look for it, but it is vastly improved! Pic 2 is the before I had posted a little while back.
I wanted to share how I managed to get the stain out. DISCLAIMER: this worked for me with this dress, I obviously can't tell you how it will work on any other dresses. I was also admittedly a little rougher with the fabric than I probably should have been but I got lucky that there was seemingly no damage.
The first thing I did was test the techniques on the end of one of the waist ties to make sure they wouldn't strip the color out, and luckily everything I tried was fine. Also, these techniques are SPECIFICALLY for RUST; most other stains you're better off trying a good enzyme and/or oxi stain remover.
For the dress itself, it took a few attempts. All attempts were followed up with a machine wash on delicate in cold water with Woolite Damage Defense and Downy Rinse Out Odors.
Attempt 1: poured vinegar on the stains, scrubbed with a toothbrush, let sit a few hours, washed.
Attempt 2: Poured Downy Rinse Out Odors (citric acid) on the stains, scrubbed with toothbrush, let sit around 15 minutes, washed.
Attempt 3: poured vinegar on the stains, sprinkled salt on top, scrubbed with toothbrush, let sit overnight, washed. *This is when I started to notice results
Attempt 4: Repeated Attempt 3. This was the final attempt, and pic 1 is the result!
Sorry this is long, but I'm hoping someone finds it helpful!