Sorry you’re right, you definitely need both to lose weight. I’m just emphasizing, if you already have a decent workout routine (ie calisthenics a couple times a week like OP), then in my experience, it’s much easier to lose weight than work it off through exercise.
I’ll frame it this way as well. People can be 50+lbs overweight (me) and still go to the gym regularly. On the flip, I find it much less common that people are obese and also have a healthy diet and food relationship.
Not sure on your fitness level, but assuming you have a little cardio base built up: Flipping tractor tyres end over end can be a really good workout. You could also do that footwork drill for cardio where you hop your feet in amd on different tires. I think searching “crossfit tires” would give some other good ideas.
Also, the rural community I grew up in had a cycling community that would go on bike rides in the morning. Are the roads around where you live safe enough from cars to cycle a bit? Maybe that could be an option. Same for jogging.
As for finding time, really you’re just going to have to choose a time and do it. Early morning workouts are the gold standard for people that work full time.