Okay... I think that both your husband and your daughter need some positive reinforcement. They both need to be "trained up" (so to speak) with some positivity. Using expressions and examples of how you would like them and yourself to behave rather than pointing out how they have failed and how don't want them to behave will work much better.
The thing is, most of her behaviors are just a kid being a carefree kid. He's withholding love and affection from her like a petulant brat because of what he perceives as her disobedience. Except he's not really laying down the groundwork of parenting.
When you go out in public, tell her beforehand what you expect from her and put it in positive and happy terms. You can set it up as behavior rules, but make them as positive as possible. So avoid, "You will not run into the parking lot; you will not touch everything in the store; you will not disobey Daddy." That's setting her up to fail and putting the ideas in her head.
1. When we're in the store, stay near Mommy and Daddy or you'll have to ride in the cart/stroller.
2. When we're in the store, keep your hands to yourself. If you don't keep your hands to yourself, we will have to leave the store.
3. When we're in the store, we're polite. We keep our voices quiet and we walk. If you don't, we will have to leave the store.
4. When we're in the parking lot, you must stay close to Mommy and Daddy for safety. If you don't, you could get hurt. [But seriously, on this one, get her in the car immediately and get her buckled in. Don't give her time to dawdle and go out into the driving area to twirl and play. Hold her hand when you're out in the parking lot and while you're unlocking the car and then put her directly into her seat and buckle her in. No excuses.]
5. Praise her when she follows directions. "I love how you're keeping your hands to yourself. Great job!" "I love how polite you are. You have wonderful manners."
6. When she doesn't follow the rules, she gets one reminder. "Don't forget... you have to keep your hands to yourself and hold Mommy's hand in the store. If you don't, then we have to leave." "I think I hear whining/shouting/crying... if that's you then we have to leave."
So you lay out the expectations you have of her from the get-go, and you also lay out the consequences. If you do this, you get to stay and you get praise. She'll learn that pretty quickly. If you don't follow directions and follow the rules, you'll get removed from the fun and lose privileges.
I know you'll probably think, "But it's annoying to have to leave and we do all of our errands as a family!" It's worth it to teach the boundaries and ground rules for good manners and positive behaviors even if it means you have to inconvenience yourself. Parenting is not convenient. There are times you're going to have to just go back later without her and without your husband. Or go just you and your daughter. Or just send your husband.
You have to learn to think of your daughter's behavior as communication. She's only 3 years old, so communicating is still really tricky for her. You have to read her behavior as language, not just her words. And you have to anticipate when the rough times of day will be, and what sorts of things trigger her behaviors.
Give her a snack before you go out. Most kids get crabby when they're hungry and tired, and you can't blame them and say they should know better. Hell, it sounds like your husband gets crabby when he's hungry and tired and HE doesn't know better than to behave like a grouchy jerk so why should your daughter be on her best behavior? He's an adult and should know better but he's not being a good example for her. He's acting like a child rather than an adult, and if he continues like this, it's not going to be pretty.