I know with kids/marriage/etc this can be hard to do but I (someone in a similar situation) find that an hour of exercise every day makes all the difference in the world.
* High intensity intervals. Do 2 minutes of light intensity cardio as warm up. Then alternate 20 seconds at top effort with 10 seconds of rest. 10 cycles of this will only take you 5 minutes. You should be out of breath by the end. * Faster version of yoga woodchopper pose[2]. Stand with feet apart and knees slightly bent. Clasp hands and raise above head. Bend knees and bring your hands down while you exhale with a loud "Ha!" Do it slow a few times for warm up, then do it fast and imagine your hands crushing a stressor at the bottom. Yell if it won't disturb the neighbors. * Tense your whole body for 20 seconds, then slowly exhale and release the tension. While tensing, imagine either beating up or successfully running away from something that is stressing you out.
[1] https://www.feministsurvivalproject.com/episodes/episode-02-... [2] https://youtube.com/watch?v=u1HjZ49xAPc
So I'm just trying to trudge along and progress when and where I can, and try my best to get my work done. I'm fortunately working from home, so no commute to add to the burnout. I've also not had a decent vacation in years. I did take two weeks off between my last job and this one, but that was mainly to catch up around the house. I tried to negotiate vacation time at this position, but they refused, which means I'm slowly accruing PTO. I'm hoping by April I'll have enough to take two weeks off and go somewhere nice.
I'm not sure I have advice that others haven't given. But it can get better. Rely on your wife or family if you can to help with the kids and house, and push them for help if need be. Find a therapist if you think that could help. Make sure to keep work and life separate as much as you can. Meaning be sure to unplug from work when you're home, and to have as few distractions from home while working. If you do find yourself getting distracted at work, see if you can walk away for a bit (even literally go for a short walk). Also, rely on your coworkers to help you. Ask, even nag, them for assistance if you get stuck. Try to make friends at work, and maybe push for fun, team-building events if that doesn't currently happen. I think showing you're trying will be big if your boss thinks productivity isn't where it should be.
Good luck!
Your guilt is false. Your kids will be fine. Your wife will be fine.
Communicate with your wife that this is what you are going to do. And continually how it is going.