You're going to need to dramatically increase your intake of sugars if you want to progress at both fast with little downtime.
You're mixing two highly energy consuming activities, the weakness and fatigue is from a lack of glycogen for the body to use. In the fitness world there is this weird demonization of carbohydrates going on at the moment, but it's based off some very shoddy rationale and very little evidence. If you were a caveman trying to conserve energy for the long winter ahead, great. But you're a modern person under stress already and now you are adding physical stress on top of it, you need the efficient energy substrate.
Muscle soreness will go down in time as you get used to the workload.
I started boxing as a light middleweight in my teens, by the time i was in my early 20s i was a cruiserweight. I didn't use any specific program, in fact i went against dogma about sports training and just did whatever felt right at the time, sometimes it was low reps and very heavy weight, sometimes it was a program that looked more like a bodybuilders.
whatever the case was, i progressed lifts, bodyweight and boxing all at the same time. I can say without a doubt i was faster as a cruiserweight than I was as a light middle, if only because of physical maturity or experience.
The only consistent "tip" i can give you is I always made sure I got plenty of sugars and salt to replace the lost fluid. Salt is the most overlooked dietary component i feel, you can't rehydrate your body properly without adequate salt, which you lose a lot of when you're active. I used to drink large amounts of fresh orange juice often with a teaspoon of salt mixed in.