Gym isn't for everyone, I tried for a long time to get fit in the gym and it was just never fun. Then I found climbing (bouldering), and fell in love with the sport. Trying out different things will help you find something that sticks.
If you do want to run a 5k by the end of the year, I'd recommend you get some kind of smart watch (I own a Garmin), which helps you track speed and especially heart rate. Turns out I'm not an aweful runner and do not hate it, I was just running way too fast for my level. Keeping your heart rate relatively low, which means running slowly, lets you run much further. Also, don't underestimate walking. Someone that walks 5k everyday is probably fitter than someone running a 5k once a week.
Foodwise I'm very far from an expert, but my personal take is that it is usually a good idea to have a mix of all the things, including carbs, fats, sugars etc. Lots of food advice looks for "evil" and then declares the rest as good, but usually it isn't that simple. Pretty hard too eat too many veggies though. And cooking is usually better than eating prepackaged stuff.
Lastly, on the burnout stuff... I'm in a similar situation right now, sport is one of the few things that actually helps me. And it isn't only us, so many people around me are very close to the edge or already beyond it. Times are really tough I think. No shame in getting help from a therapist, or finding help groups or something like that.