Since having become a full-time game developer since June, i've been the fittest i've ever been in my life. I personally aim to do between 5 to 10 hours of high intensity Cardio a week. Ive now let myself become addicted to cycling, it is really great fun, keeps my mental health in great shape and also is constantly improving my body. For me, it's all about having more than just one focus. Stress seems to dissipate for me when I can switch gears and focus on my training rather than purely on game dev every minute of every day. There have been times when I have let development consume me, and whilst I'll generally be incredibly productive for a short period, it is not at all sustainable and just ends up burning me down in every possible way.
The simple solution to your problem is to start mixing a bit of activity into your routine. It doesn't take much before it starts having a significant impact. If you start by aiming to do an hour or two of exercise a week this can really help.
Regarding what others have said, just moving around during the day helps a lot as well. I tend to get up and pace around whenever thinking about a problem, but equally, i'll sometimes constantly be going back and forth getting drinks from the kitchen when stuck on a difficult problem. This may sound counter-productive, but most of the time i'm moving, I am thinking and planning inside my head in much the same way i would be if I were sat at my desk.
If it's hard for you to mix in things like exercise, then to make it easier, its just a matter of replacing certain tasks with slightly more active versions, for example walking to the shop instead of driving. Taking a slightly longer route when the opportunity comes up. When moving around, try and go a bit faster. It sounds silly, but running through your house/flat rather than walking will not only make you feel better, but also give your body a bit of opportunity to stretch and get your heart going.