Firstly, I’d say to take a deep breath. Everything is going to be fine.
Yes, your nutrients are high, but you’ve already done the first and most important part which is identifying it. And you’ve already started taking steps to bring them down. And if it’s any consolation, I have seen a couple dozen tanks over the years with nitrates between 500-800ppm (mostly managed by the same people). I was brought in to help fix these, and doing some very large water changes definitely helps.
If you don’t want to add any chemicals, and I don’t think it’s necessary unless you want to, then I would recommend the following.
1) Do a couple of water changes that are 20-40% each. One big water change is much better than two smaller water changes of equal volume. Keep this in mind when you plan.
2) Wait 48 hours between these big water changes.
3) When you’re doing the water changes, try to siphon the sandbed as much as possible. If it’s dirty, all the water from the bottom of the tank will likely be dark brown or gray. It’s OK if you don’t get all of it on the first big water change. Any little bit you do get will be beneficial to removing nutrients.
4) Heat up the New saltwater prior to doing to water change.
5) Double and triple check the salinity and temperature of both the tank and the new saltwater.
6) and lastly, stop feeding so much! Lol, half-joking. But it is important to identify how they got that high to prevent this from all happening again.
And once again, you’ll be fine! Keep up the good work, and keep us informed of how everything is going! Like the story of the turtle and the hare, ‘slow and steady’ wins the race.