The first thing I would do is speak to the care home manager and find out how your mum is when you are not around. Some people can play up when loved ones are around but the home then tells you that this is not the behaviour they see when she is there on her own. If this is the case then it can determine how you arrive and how you leave. For example, you can quietly arrive with no big fuss and you can leave in the same manner. For example, no big goodbyes, dad can get up and say he's going to the bathroom, then leave and wait for you outside. You can get up and say you're returning the teacups to the kitchen, go find the carer and tell them you are leaving so they can go talk to your mum. It could also be that if she is ok when you're not around that you consider visiting less often until she is well and truly settled. My mother after a while was more institutionalised and became more interested in what she was missing than spending dedicated time with us. We still visited regularly but tended to socialise with everyone in the lounge rather that going and sitting in her room with her on her own. This made sense because everyone else had become her extended family and we had to fit in with everyone else too. It wasn't just us, but other visitors all joined in too and this had the benefit that visitors were for everyone and didn't leave the ones with no visitors excluded. Same if we took cakes or biscuits or magazines in, they were for everyone and this worked well too.
Rather than using the teddy bear, you could try getting your mum a baby doll and a blanket. Introduce it by you sitting holding it and watch for interest. Ask her if she would like to hold the baby, then maybe she could watch it for you whilst you nip to the shops...then take your leave. Doll therapy can be very useful and can reduce agitation but also provide companionship and give a sense of responsibility too.
Hope this helps,
Fiona