When you lower your carbs are you replacing the calories/volume of food with more fat and protein? That may be the main reason that lower carb meals leave you more sated, more so than the lack of carbs.
Working out with intensity (think sprinting, weight training) uses glycogen. Emptying your glycogen tanks improves your ability to metabolize carbs since your muscles and liver will want to restock glycogen.
P.S. Sorry to hear about your rough go of things recently. And welcome to the forum.