Good advice above.
Yes on the cook things separately, being very careful not to overload the pan. By overload, I mean that the pan temperature MUST NOT drop too much or you change from stir fry to steam/boil.
Some items benefit from other cooking methods prior to the stirfry. Brocolli and cauliflower are tricky to tenderize without some moisture. You can blanch as Ed sagely mentioned or you can stir fry it a bit, add a bit of water and cover in your wok for a few minutes. Let the pan come back up to temp before cooking the next item.
I'd also get rid of the non-stick wok. It contributes to some of your problems. More on that later.
For your listed ingredients, here is what I would do.
Prep all vegies, meats and sauces. Marinate the shrimp in corn starch, shaoxing wine and soy sauce, about a tablespoon of each liquid and 1/2 tablespoon corn starch, assuming 4-6 oz of shrimp.
Heat the pan. Add a bit of oil.
Cook the shrimp first. Spread them evenly in the pan and let them sit about 1 minute. Then stir fry 1 minute more. Remove and set aside.
Bring pan back to temp and add a bit of oil. Add broccoli, toss a few times and add a bit of ginger and garlic. Toss a few more times to distribute seasoning. Add no more than 1/4 cup water or maybe rice wine. Cover and let steam 2-3 minutes. Rmove brocollli, set aside; dump any liquid.
Repeat for cauliflower or cook at the same time if it's not too much (1 1/2 cups is about the limit for these types of vegies.
Reheat pan, add oil. Add onions. Distribute evenly and let them sit for a minute or so. YThis should give them the edge sear and color you see in a good stir fry. ou can add some more garlic and or ginger at this point. Chile garlic sauce would also be good. Toss together a few times and remove, set aside.
Repeat onion step with peppers and mushrooms separately. Onnions, peppers, mushrooms will release a lot of liquid if given the opportunity. HIGH temp and cooking them alone helps prevent that liquid from building up and becoming a boil. Mushrooms will take probably 2 minutes before stirring to pick up the right sear.
Add everything back in stir/toss to combine and add the premixed sauce ingredients. Stir the sauce before adding to ensure that the corn starch is in suspension. Heat and stir until the sauce thickens. Depending on the quantity of your ingredients, you made need more or less of the sauce I describe below.
Basic Chinese Brown Sauce
3/4 cup beef stock, low sodium
1 tablespoon soy sauce
1 tablespoon oyster sauce
1 teaspoon corn starch
some ground white pepper
a pinch or two of sugar (optional)