Being tired from work is real. I'm dealing with that currently. Surprisingly though, when I was working on my feet 9+ hours a day and commuting, exercise actually helped. I realized it was a mile home from the transit station, so for a month I ran it. I had to traverse it anyway. I just swapped into gym clothes when I left work. For such a small change, it did wonders for stress relief and health.
It doesn't have to be. Eat a little leaner, and move a little more. That's really it. Doing some exercise consistently is a million times better than being overwhelmed and doing none.
I have great news. Legs are amazing. I can't speak for everyone, but mine build up fast when I try. Quadriceps are huge muscles! Yeah, that wobble does not sound great. Definitely pay attention to your form and posture, and listen to your joints.
Thankfully, losing some weight and generally improving overall health should help with this, as well as strengthening your legs. You get to attack with this with a two edged sword. Less weight
Great self awareness. With the above in mind, I urge you to really consider if muscle gain is more important to you right now than weight loss. Lower weight would give those legs an easier time. Frankly, it's easier to stabilize and carry a lighter person. That wobble sounds concerning. I don't know the severity. You may want to drop a good 10lbs or more before focusing as much on muscle. You be the judge.
Is there any way I can help? I'm no expert, and no Olympian, but I enjoy the topic, and I'd be happy to share what I've learned and what's worked for me.
You need a basic fitness strategy, and from there, a plan.
Here are some options, with simplified ideas:
Bulk
Focus: Muscle gain
Main moves: Weight lifting. Resistance training. Squats + bench press + back and shoulders. Potentially work towards highest weight lifted
Diet*: More protein. Lean protein. Healthy otherwise. Highest total calories
Note: Bulk smart. Do not add more fat. (You will also burn plenty of fat if you do it right, as with any of these.)
Balanced (Recomposition, or 'Recomp')
Focus: Balanced. Simultaneous muscle gain and weight loss.
Main moves: Lifting + Cardio (balanced, it's in the name).
Diet*: Overall healthy. Medium total calories
Note: Safe choice. Mentally easiest for some, and most approachable workouts. Can be slower. I've done it. It was rewarding, but took time.
Cut
Focus: Fat loss
Main moves: Cardio. Running, swimming, cycling. Potentially lower weight lifted, for more repetitions. Potentially more self-supported movements like pushups, sit-ups, and chair raises.
Diet*: Lowest total calories
Note: You're going for calorie burn per hour, and/or duration. Keep moving. I don't think I've ever purely done this.
* Some will advise low carb, some will say keto. I can't quite say. I know I've read and played with the MyFitnessPal macro calculators, and generally more protein is good.
That's all a bit simplified. If you've ever heard "bulk then cut", I know it sounds like frat boy nonsense, but my experience says it holds some truth. You're going to do a mix, not matter what, but having a target helps. Which appeals to you?