How can I improve my fried rice?
This is based on what I was taught by a Chinese cook when I worked in his restaurant at age seventeen. Any compliments should be directed at old Tommy Wu.  Any complaints may be due to my imperfect memory. His process was both similar and different in some respects from yours.


Use day-old cooked cold white rice.  Spending the night refrigerated will make it drier and easier to break up sticky clumps.
Fry your scrambled eggs first and set them aside.
Use a well-seasoned wok or pan with very little oil.  Do not add broccoli, onions, or other vegetables at this point. They will be added after the plain brown fried rice is done to turn it into a specific dish.
Dump the cold rice into a cold wok or pan and add your basic flavors on top of it.  I remember the proportions and ingredients as being roughly 3-4 gallons of rice, 2 cups of regular soy sauce, one cup of thick black soy sauce, 1/8 cup of salt. These are rough guesses on my part since we did not measure anything precisely.  The "cups" were actually big metal ladles and the salt was some amount at the bottom of the ladle. You would have to experiment to proportion it down for home quantities and to match your taste since you want to limit your sodium.
Crank up the heat and get ready for a workout.  A commercial wok makes an extraordinary amount of heat. I might use a cast iron skillet with high sides or a dutch oven at home. Either way the stove should be cranked up as high as it will go. The main thing that people do wrong when frying rice is that they are afraid of getting it too hot and burning it. You want to have your pan super hot but to keep the rice moving around to prevent it from burning.  That is one reason why you should use very little or even no oil since it would smoke in this circumstance.
Assuming you are right handed, take a metal spatula in your right hand and a metal ladle in your left. Use the spatula to scrape every inch of the pan repeatedly into the ladle which is a backstop. This is done with both arms moving in an upward "tossing" motion. If you have the heat high enough, you will have to work quickly and constantly for several minutes scraping and tossing. Don't neglect to scrape the far side of the pan or it will burn.
Every thirty seconds or so, hold the spatula vertically into the center of the upward-facing ladle and hammer them down together over a variety of spots in the pan. This gets the rice hotter and really gets the browning effect you want. Only do this stamping for ten seconds or so and immediately get back to scraping or it will start to burn.
After about five minutes, it should be steaming nicely, the browning will look cooked rather than merely dyed by soy, and your arms will feel like they want to fall off.  Get the fried rice out of the pan quickly into a large bowl and then stir in your scrambled eggs. Since you are adding them at the end, they should not break up nearly as much as they were for you.
Now you can eat it as plain fried rice or use it as a base for a fancier dish with meat or vegetables.  This can be kept and reheated for a day or two if you want, as long as it is will covered.  It is best kept warm (140 degrees?) in a steamer to keep it from drying out.  If I remember right, we would cook the meats, vegetables, and mushrooms in a brown soy-based sauce first, then add the warm brown rice to that pan when the meat and veggies were done.  That keeps you from cooking the rice twice and lets you cook those components in a way that is more suitable for them.