Stargirley, you sound really frustrated, angry, and exhausted. It also sounds like you tried really hard to be calm and reasonable (telling her she didn't have to wear her coat) ... and I wonder if you also felt hurt (when she yelled that you don't care about her).
I remember being pregnant and having to care for my older child: it sucked, alot of the time. I'm sorry it's hard for you, too. Motherhood is tough when you're NOT fighting hormones and fatigue, and pregnancy makes it at least doubly so.
I wonder if I might be able to help by suggesting a few things that MIGHT be going on from your daughter's perspective. I could be altogether wrong about all of this, but maybe if I hit on something, you'll see a new possibility for interacting with your daughter that feels better for your whole family. The way I understand it, all communication is really about trying to get our needs met ... so I'm gonna suggest some possible needs that might be underlying your daughter's actions.
First, frankly, I can understand her not wanting to go back out into the rain once she'd gotten home, especially after being at the babysitters for what seemed to her to be a long long time. I can imagine she might want to stay home to fulfill her need for rest, comfort, playtime with you. I know ^I^ don't want to go back out once I get home from a long day, if I can avoid it. (Please understand that I'm not saying you were wrong to take her home to go potty, but just that maybe this could be part of what was working against you last night.)
And I don't think a five-year-old has the concept yet of weighing the importance of things that don't have direct impact on her: did you need things from the store that prevented you from making her anything to eat for dinner? If you didn't, then I can COMPLETELY see that she wouldn't see your need to go out as more important than her desire/need to stay in, or even relevant. And even if you did have to delay making dinner, I'm not sure that's important enough to her -- she probably knew there was something in the house so she wouldn't starve. And maybe she wasn't even hungry, anyway. So what needs of hers could have been met by going to the grocery? I can't think of any offhand, and I can think of several reasons she'd want to stay home.
Maybe she thought that if you stayed home, she'd have even more fun with you ... does she have any fun when you go to the grocery, or is she crabby in the cart, like my boys sometimes are? Do you have any grocery games to play?
Once I get into battle mode with my kid, it's hard as hell to shift out of it, for either of us. I wonder if it's the same for you? And my boy has the benefit of stubborn genes from both of his parents, as it sounds like maybe yours does, too. Plus the influence of your fiance in her life.
I wonder, is your daughter maybe feeling like she's being displaced from the center of your life, what with a new step-dad AND a new baby? Sibling rivalry is tough. I remember how I felt growing up with my little brother (and I can reconstruct my feelings based on looking at lots of pictures of me -- I was only three when he was born): I would have liked to kill him! And blended families are challenging, as you know, even for adults with lots more psychological, emotional, and mental resources and experiences than a kid has.
Anyway, even if you're giving her what YOU see as "plenty of one-on-one attention," I wonder if she feels that it's plenty, and/or if she knows that it's decreasing or about to be decreasing? Think about it: when you're five, you're really REALLY present-time-oriented. The whole of your remembered life consists of just a few months, really -- so being separate from the person you love most even for a few hours seems HUGE, not to be mended by an hour or two at a time.
Would you consider, when you're not in the midst of a struggle with her, asking her how she feels about her time with you? Would you consider asking her how you can have better times together instead of feeling tired and crabby with each other so much? When you say "we do our own thing after the babysitter," you might want to consider whether going to the store is "our" thing or just yours, and she sometimes/usually goes along with it but really has no fun.
And maybe the tired part is a big key here: is she really really tired after being at the sitters? (And we know you're tired after work, that's a given, right?)
Do you know the old HALT thing? When you're Hungry, Angry, Lonely, or Tired, you need to be especially attentive to being gentle with yourself and those around you ... and attend to those physical needs ASAP! The same would be true for your daughter. (My boys are BEARS when they're hungry and tired! Me, too, frankly. And add hormonal to the bunch, and WATCH OUT!)
Please PLEASE hear that I'm not blaming you or trying to make you feel wrong in any way. I'm just offering you a possible view into your daughter's world. I may be off-base, but I think maybe I've got some good possibilities for you to think about.
And I'm not suggesting that your daughter's needs are more important than yours (even if SHE appears to think so!). What I'm suggesting is, as fully as you can recognize her feelings and needs, it will help you work WITH her to find ways to meet both your needs. You'd be amazed at the ideas even little kids can come up with to meet both your needs.
Now I'm gonna tread on REALLY thin ice here, but I feel I need to say it. I'm really glad you didn't let your fiance do the "wait til he comes home" thing. But I'm not sure that she sees taking all her toys away as a reasonable alternative. Remember that, at that age, play is their work. Boredom will not create better behavior. Maybe she can still play with her toys when she's in the room with you, or something ... but if she really has nothing to play with, that's gonna make more trouble, in my opinion. And if she feels punished for actions that, in her mind, are perfectly reasonable attempts to get her needs met, then that's got to be confusing and difficult for her.
Now speaking of kids' needs, I've gotta turn off the computer and attend to my own boys. I'd apologize for my wordiness, but as you can see, this is a topic that feels really important to me. And because I don't practice this NVC all that well, when I type out suggestions to others, it helps ME, maybe even more than you.
Thanks for reading this far.