If you think it's hard to deal with this now while you're pregnant, wait until you're exhausted with a new baby. Her behavior will be worse.
You have mentioned here that she isn't the best talker and is repeating your remarks to her. I urge you to call the ped and ask for an evaluation with a speech therapist. Even if her enunciation of sounds/words is age appropriate, a speech therapist will check her language understanding. Both receptive (what she understands) and expressive (being able to express herself) are very important.
She is frustrated, as are most 3 years olds. But frustration turns more hard core when there are language or speech problems. Throw in a new baby on the way, and it's a mix for major upheaval. Get a grip on it now by getting her evaluated and some help on the way.
Meanwhile, when she starts this stuff, get your face right up to hers and say "Use your words". In front of her, wait for her to try to tell you. If she's screaming, quietly say "I can't understand you when you scream." If she still won't try to talk to you, THEN put her in her room. ANY toys that she throws at the door need to be gathered by you and "put in time out". Tell her that the toys are in time out because she threw them. Tell her that toys don't like being thrown. Put them in a box and put them out of her sight. Pretty soon, there will be NO toys in her room. She will have to earn them back with good behavior. That's the point when you can say "Remember, I told you that if you want to earn your toys back, you can't scream at mommy." At the end of the day when she has held herself back by listening to your reminder of good behavior, give her 1 or 2 toys back. She probably will not throw them again.
I would not put her in time out anywhere other than her room or the car when you are away from home. Strapping her in her carseat and standing outside of the car "reading a book" ignoring her is the best way to deal with struggles away from home.
Don't take away dessert as a punishment. Give dessert as a reward. Give TV as a reward. "You listened to mommy's words today and did what mommy asked. You may watch one TV show/have dessert/etc." That way she sees positivity rather than negativity.
It's okay for her to cry herself to sleep. Until she knows you mean business and she is not allowed to act like this, it's best that she is in her room alone and working this out on her own. Too many different consequences, threats, etc as you try to figure out her "currency" just makes her try more different things to freak her tired mommy out. Don't let it. When she won't stop despite your warning, you can say to her "You must be tired, so you need to rest in your room." Don't let her come out until she is behaving, and when you DO let her out of her room, make sure if she threw her toys, you take care of that.
It's not dumb to feel awful or embarrassed about our kids having tantrums in front of the public. What's important is that you don't cave into what your child wants and that you remove her from the public so that they don't have to listen to it. Even if it means that you walk away from your cart, get her to the car and strap her up and let her cry. When she is finally done, say in a strict tone to her "Are you ready to try again and do what mommy has asked you to do?" Then take her back in the store and continue shopping. Even if you have to do it 10 times (shopping boot camp), you need to show her that you mean business and that you won't let her run roughshod over you.
It will get better if you are 100% consistent.
Good luck!