Such a very difficult time for you, Merename.
I encourage you to have frequent and open conversation with the carers and other staff at the care home. It's a good way to start to form relationships with them. I would definitely not hesitate to ring them anytime you want an update or are feeling anxious.
While it can be very hard to do, in some cases, it is better to decrease frequency and/or duration of visits during the settling in period. Again, I'd talk to the staff and see what guidance and advice they can offer you.
As regards the clinging, that's very distressing. If possible, can you time your visits so that you will leave just as something (a meal or an activity) is beginning, so that it makes a more "natural" end to your visit? Or ask one of the staff in advance to help you, by providing distraction in the form of a cup of tea or something similar.
If your leaving agitates your mum, then you may want to keep goodbyes very short and sweet (like when leaving a reluctant child at school the first time, the quicker you go, the better), or possibly not even announce you're leaving--you're just popping out to the toilet or something. That approach depends, of course, on her memory.
I also found, with my mother, that saying "I will see you Tuesday" or even "I will see you tomorrow" agitated her, as she couldn't keep track of days and got confused and upset trying to do so. Now I always offer a very cheery "see you soon!" and a wave and big smile as I leave, which works better.
And I had to learn to take what my mother told me about the care home with a grain of salt--another good reason to talk to the carers and staff over the phone. If you believed my mother, she was alone in her room with no food, none of her medications, no sight of any other people for days, and bored and miserable. I didn't quite believe it, of course, but it was a real relief to hear the staff tell me that she joined in with activities and made friends and so on. I also sometimes would visit where I just observed, from where my mother couldn't see me, so I could see for myself how she was. I don't know if you will find that helpful, but it's a thought.
My mother is a care home success story. She settled better, and faster, than anyone had hoped, and has really thrived. Maybe most surprisingly to me, she was able to adapt to the care home routine--I wouldn't have believed it until it happened. I only wish I had been able to move her into care sooner (as her prior situation was so poor). I know it doesn't work out for everyone, and we can't predict the future, but I do hope this situation will work for your mother.
Very best wishes to you. Remember, it will take time for you to adjust to the new situation, as well as your mum. Please try to get some rest and look after yourself (much easier said than done, of course).