Oh, goodness. This is a good example of what not to do with a dog that's fresh from the shelter (FFS)...
She's been living in a cage in a shelter for...a while. She was insecure last week. She's feeling more secure, so all the pent up anxiety, energy, and even play that she couldn't express in the shelter are going to come out. They need to come out so that she can start to settle down. You still don't even know who this dog is--it takes more then two weeks for all the FFS energy to dissipate. Shelters are stress-filled places for most dogs!
No FFS dog should have run of the house when I'm not supervising them. My fosters don't even earn that after a month with me! That's inviting trouble. They don't know you, or know this place, much less have any inkling of your "rules." They might as well have been dropped on Mars and have Martians talking at them--it's all alien to them. Many of them haven't even been inside a house, ever. She has no clue how to distinguish your things from her things right now -- it's not her fault.
The first couple week, you should have NO expectations of training. Let them instead learn to trust you, learn the rhythm of the house, and its sights and smells. To that end, the advice you were given to keep the dog on a leash next to you is spot on! Tie the leash to your belt. I even put a special low hook in my kitchen so that I can have my foster dog leashed close-by when I'm cooking--they can smell, and watch, and experience.
My goal with any FFS dog is to set the dog up to always be in the right place, doing the right thing -- that way they experience lots and lots of positive reinforcement (and there's no need for corrections--if they screw up, it's on me because my job was to set them up to succeed during this critical time).
What happens in the 2 weeks of "leash time" is the dog starts to see you as the source of all things good--esp. companionship. You start bond. They start to trust you and respect you. THEN you have a good foundation for training.
You can't expect a FFS dog to magically not pull on a leash when you go for a walk. Dogs aren't born with leash manners. They're acquired through good training.
Can you sign up for a good obedience course? I think you and the dog would benefit tremendously. You'll learn terrific skills and communication techniques. You'll learn how to show the dog the behaviors you want. It will deepen your bond.
Crates are incredibly important too. Please properly crate train though -- don't just stick a FFS dog in the box. Make it a good, safe place. Feed in the crate. Give lots of treats in the crate with the door open at first. Give the dog a sanctuary, so when you have to leave the dog, or need to take a break from the dog, it won't be traumatic.