I agree that you need to leave her with someone you trust, telling her bye and when you'll return. The more often you do this the more quickly she'll learn that you are returning and thus not be quite as upset.
Another idea, while at home, is to set her up with playthings near you so that she can see you as you do other things.
Her clinging does seem excessive to me. Has she had any negative experiences that frightened her?
Perhaps without knowing it she is picking up on your anxiety and frustration. Because she doesn't know why you're feeling and/or acting that way she feels more insecure and clings even more.
I've heard the suggestion to handle clinging children by slowly increasing the amount of time that she is without you. Go out for 1/2 first reassuring her that you will be back in 30 minutes. Gradually increase the amount of time you are away from her.
When you're at home and she clings, calmly set her down, give her something to play with and as difficult as it is do not pick her up for 10 minutes and then gradually increase the time. Put a baby gate on your room or the bathroom so that she cans see you but not get to you. Perhaps play some calming or cheerful music while you're focused elsewhere.
Yes, this is the age at which babies are trying new things. They do not have a sense of permanency. If they don't see you, you don't exist for them. This is why peek a boo is a good game to play with them. I suspect that sensitive babies get overwhelmed by the new things that they learn at this age and when they cannot see you or you are understandably angry or frustrated they become hysterical. Getting thru this stage takes lots of patience. You do need time to yourself. And she needs to learn how to be without you starting with short periods of time increasing to longer times.
I've read children's books to my daughter and grandchildren that tell a story of Mom and Dad going out and having a babysitter. Perhaps reading books to her would help even tho she's not very verbal. Kids understand more than they can speak. Also by reading these stories of other kids or even calmly telling them stories will take some of the tension out of the situation.
It takes time but calmly tell your baby that you are going to have a shower now and give her some toys and leave her with her dad might help. Dad should then focus entirely on her, helping her to play, reassuring her, reminding her that you will be out in a little bit. She will scream at first but I think that if you keep doing this and are able to not respond to her scream she will eventually realize that you will always be there even when you're busy doing something else.
It's especially important to respond to her clinginess and hysterics in a calm reassuring manner. Expressing anger and frustration makes her feel less secure. She is totally dependent on you and she doesn't know what you will do when you're upset. If you put her down in a harsh manner because you've lost patience she is probably responding to the harsh manner as well as not being able to have you hold her.
It's tough being a parent and it's nearly impossible to have private time when your baby is so young. But they do grow up and before you know it they don't want to have much of anything to do with you. I wish you well!