Exercise doesn't really take that much time, you just have to prioritize it and select activities that will give you the biggest payback for the time put it.
I recently completed my first sprint triathlon (it was 1/2 mile swim, 19 mile ride, 3.6 mile run), and I'm in really super shape from training for it! I've always been a healthy weight, but I'm talking starting to see ab muscles now. Respectable cardiovascular endurance. 56 bpm resting HR. My arms look rad from all the swimming. I look and feel great. But do you know how much I trained? Not that much. Like 30-60 minutes 4-5 days a week. That's just how much activity you are normally supposed to get. Also, I have a very supportive partner (but he also works full time), and we have two kids that we don't leave at home alone yet (ages 6 & 8). And the kids have activities- scouts, occasional lessons, school events, weekend Chinese school. (I also attend Chinese school!) Also, I watch a good bit of TV- several sitcoms, a couple of dramas, probably a few hours a week. I also waste time on my phone, fb, ig, texting my BFF or messaging my almost-BFF training buddy. I do work fewer hours than you, but my point is you probably have the time.
168 hours in a week... let's say 50 for work, 10 for commuting (if you have a long commute), 56 for sleeping 8 hrs a night, 7 cooking, 17.5 eating, 7 cleaning (that's a lot of time for grooming yourself and cleaning your home, IMO). That leaves 20.5 hours for other pursuits. That's a lot of time! You could sit down and watch TV, or you could get active. Honestly, with 20 hours a week you can do both and still have time to spare.
Map out your weekly schedule someplace (I like excel) and see where you have time that's unaccounted for. There is probably time to get fit.
Low on energy? Get some proper nutrition in so you know you are physically capable and do it anyway. It's mostly mental. You'll feel great after a workout!