Your husband is 100% correct. You and your family do not exist to please the neighbours. Your family has needs and your neighbour has been expressing preferences. You have been treating your neighbour's preferences as being more important than your children's needs.
This is wrong.
Your children will have a very highly developed sense of what is fair and right. Your children will also be extremely tuned in to your stress levels. No wonder they are showing signs of distress.
The reason you are so stressed about all this is because you are experiencing an internal battle. On the one hand you have ingrained 'lessons' learned all your life to fit in and comply and not rock the boat. You want to get on with people and be liked.
This is classic for an undiagnosed autistic woman. Women and girls are 'trained' to behave and comply and the dreaminess and shyness and rigid following of instructions typical of autism can be seen as just being 'girly' and well-behaved and doing as you're told. I know because that is how I managed to get to my 50s before realising I was autistic. Women are often great at observing how others behave and then mimic that behaviour to fit in without even realising they are doing it. Not fitting in becomes a huge source of our self-esteem and when it doesn't work we get very anxious and distressed. It is one of the reasons we become people-pleasers and put others' convenience and needs ahead of our own.
At this point if you put your children's needs ahead of your own that will help get rid of the internal struggle between wanting to fit in and feeling annoyed with yourself for giving in to this woman because your children's needs are 100% more important than this woman next door's whims. The 'group' you need to fit in with here is your family group and not the neighbour group.
You don't want your children growing up thinking that they should put their well being last and prioritise stranger's whims. Otherwise they could be heading for an even tougher life ahead of them than they will already be facing. You don't want your kids to be struggling with similar internal issues when they meet up with nasty selfish people you want them to be able to protect themselves against being exploited. This woman has exploited your good nature and instinct to co-operate.
Your children deserve to have a home where they can relax and be themselves without feeling they are being scrutinised. Keep track of what this woman says (on paper and/or if you can manage it with a hidden recorder - hoodies with front pockets are very good for this) and does and if she continues threaten her that you will contact the police with a charge of harrassment.
You are entitled to live in peaceful enjoyment of your own home and that includes normal (for you) family noise and if she can't live harmoniously with that then she will have to move.
I would also remind her that if she starts up a neighbour dispute that you won't be moving and that a neighbour dispute would have to be declared to potential buyers of her property and would bring down the price of her property and so it is in her own financial interests to stay on good terms with you.
I hope this has been helpful.