Well, this is just my personal opinion, but i will do my best.
I noted, as you have, that many of the concerns you mention stem from differences in how you and your lady relate to your respective families. Neither of you is wrong in how you relate, it's just different. And when we care about someone in a relationship, we need to try to address their concerns from their perspective. In other words, you can't expect her to just stop caring or listening to what her daughters think. And they probably assume you care as much what your sons think. I think you're going to have to try to treat the lady and her daughters almost as a unit.
I think if she is spending more than she is earning, that is a valid concern, and i can see your concern is just because you care for her! You don't want her to be deprived of anything, you just want her to be financially secure and independent. That's being on her side. I think that is wonderful! So with this i think you just need to communicate better with her and with her daughters. It IS still her decision how she spends, so you must respect that, even if she chooses foolishly. But i think if you let her and her daughters know where you're coming from and that it is just because you are concerned for her, that might go better. Perhaps with the right approach, once they see what might happen to their mom if she overspends, you could enlist their help in talking to her about her spending.
You also mentioned their asking to see your family. On this, and again this is just my opinion, but i think you should give in, and at least ASK your family and some friends to meet with her. She is obviously very important to you! And meeting them is obviously important to her and her daughters. They are a close family, and this may also be a cultural difference... to meet you and yet not meet the other important people in your life may feel like they are only seeing a part of you. i can see how they might feel like this is a lack of commitment and openness on your part. Whether or not you quite understand their need, they still have it. And i think arranging a meeting or two or three would do an awful lot in gaining their trust and belief that you really do care about her. Ask your sons. Leave it up to them. Your sons are grown up... they can each start deciding for themselves. Just tell them that you are dating a wonderful lady who has become important to you, and would they come round and say hello? You don't have to make it a big deal, or only about meeting her. Perhaps on the next small holiday you could plan a dinner. The longer it goes without meeting, the more a big deal it becomes. You don't even need to phrase it that the purpose is to meet her.
As far as the kitchen goes... has she herself complained? Doesn she enjoy your cooking? I would let that be the answer. Of course... it is very romantic to cook together!