Don't take her shopping, grandma. She's playing you but good. Your 14 year old is old enough to watch her at home after school while you run out to do errands. You CANNOT hurt your back over this child. If you do, you can't take care of ANYONE in your family.
If for some reason you have to take her out, make sure she has had a nap and a snack beforehand, and take your 14 year old with you. If she pulls this stuff, the 14 year old needs to grab her up but good and you two march her out to the car and let the 14 year old strap her into her carseat. Then stand outside the car while she cries. Once she stops crying, ask her "Are you ready to behave?" Then take her back in and try again.
Quite frankly, I don't think you should take her anywhere that doesn't have a cart to strap her in.
I hope that you are sending her to her room at home when she is misbehaving. If taking her toys away and timeouts don't work, put her in her room for at least 10 minutes. Do it all day long until she gets the message that you aren't going to give in and let her run your house. The other thing you can do is to put her toys in timeout AS WELL as putting her in her room. When she starts losing so many of her toys, and when she no longer has you but is alone when she does stuff she's not supposed to do, she will start to learn.
Right now she is in the middle of the terrible two's and her brain will have to develop into understanding that she has to obey you. The ONLY way she will learn this is if you are 100% consistent. That means all the time, grandma. It might drive you bonkers, putting her in her room all the time, but you must do it if you want her to grow into obeying you. You need to get your 14 year old into the groove with this. Have a word that you use with her so that there's no conversation, and she just picks the little girl up and deposits her in her room. If the child keeps opening the door and coming out, put a latch way up on the outside of the door to keep her from coming out. And don't talk to her from outside the door. She needs to feel alone. If she wants to be with you, she has to behave. She will learn if you stick with it.
Good luck!