goodness, you are walking on eggshells aren't you? No wonder you're getting depressed.
My DF phoned today & said that I shouldn't argue with her as she is my mum. He would never have argued with his mum
This is how things used to be in repressive families but it's a Victorian view. Things have (thank god) evolved and it's okay to have a civil disagreement and to work out what's feasible for you, what's feasible for the other person and to meet half-way and respect each other. Even with your parents!
Your father probably didn't do things the way his father expected in some aspects and you don't have to do things the way your father does.
I think you need to start putting boundaries in place and not dancing around her or your father's pressure. It's okay to say 'yes, you embarassed me by bringing clothes to the restaurant'. She won't like it, she will sulk and will sulk more as you keep on doing it but when someone starts setting (appropriate and reasonable) boundaries almost everyone finally comes round and accepts that even if they don't like it.
Your main problem here is that for a time you will feel bad when she starts the emotional stuff. But living with her acting like this isn't doing you any good.
(I'm not advocating cutting her off btw, not at all. But I do think you need to put up firmer barriers and stop talking to her about personal things. It'd be nice to be able to talk to your mother but an awful lot of people find they can't especially as they become adults).