My husband as previously stated is very protective of her as we have 3 boys but only 1 daughter and I think he’s he still views her as his little girl
If your husband is a sexist who treats his daughter differently to his sons and won't respect her as an adult then tell him he needs to grow up before he loses his relationship with her. You say you think she's more concerned about her dad's behaviour - yes, if he is boorish, liable to make even a faintly sexist comment about 'his little girl growing up' or similar, all the while having his arm slung possessively around your shoulders with the odd 'quick peck on the lips when appropriate' thrown in (!! it's never appropriate surely?!) - then yes he will look like an idiot in front of her boyfriend's parents, and she will be embarrassed.
I think she’s worried that we do speak with a northern accent and aren’t as posh as his family are...I also think that she’s concerned that it may come up in conversation that she is having to fund herself through uni as we can’t afford to help her financially
But she hasn't alluded to any of that, has she? She's been quite clear- she'd like you to not 'joke around' with these people that you've never met before (which sounds like it means possibly the odd ill-timed dig about her, the 'little girl' (boak), she'd like you to not act immaturely, especially PDAs - which everyone on here has pretty much said are always inappropriate. In short, she'd like you to come over well, and simply get to know them. She hasn't asked you to not mention money or speak nicely. So don't deflect it - don't do her down. Nothing you have actually said that she has said indicates that she is at all embarrassed about her background. Merely the way you behave. So once again, I would say don't get chippy, listen to her.
It sounds like her dad is on very thin ice.