I wish I had really useful, relatable advice, but I know that I learn languages quite differently from many people, so no guarantees anything I do will work for anyone else at the same speed…
How about this? Do you have a list of these?
If I have a list, I might be able to give you an idea of how to give them an overarching structure instead of memorising them one by one. (Yes, Genki and all other textbooks typically organise grammar points in a decent way, but the fact that they treat these things as ‘points’ to be learnt individually instead of things to be compared and understood together is often counterproductive in my opinion.)
The issue is that even if these alone allow you to break sentences down, you’ll need vocabulary in order to understand what’s going on. There are usually relatively few kanji on the N5; I don’t know how things are for the N4, but there generally won’t be a ton either. I think knowing words is going to be more important. It’ll probably help you if you can run through the Genki passages quickly. Do you have translations available? That should help you get through them faster. The key thing to focus on when working with translations is understanding how Japanese words literally come together to give you the meaning given in the translation. Of course, direct translations don’t work very well for Japanese beyond a certain point because English and Japanese express things quite differently, but you can at least try to see a connection. That should help you understand Japanese at a more fundamental level, which should (at least in my experience) make it easier for you to remember things and to learn new ones, because they make more sense.
As for how to learn words… depends on you really, but for me, I do a lot of visualising, reading aloud and writing, and I try to combine them all. My goal is to create a vivid, even emotional connection between symbols, sounds and meaning, and to make that connection intuitive (i.e. if I can find a shorter mnemonic to ‘explain’ what I’m seeing that makes me go, ‘Of course, so that’s why it means what it means!’, I take it). If visualising is hard for you though – which is fairly common here, honestly – then you’ll have to look for other ways to make what you’re learning interesting or memorable, maybe by noticing a funny sound in a word or a similarity to something you already know. Whatever you do, I think that a good strategy is to look for connections with things you already know, even if they’re not clearly connected to Japanese, because it’s easier for your brain to use something you already know as a stepping stone to learn and recall something else. That’s what I would suggest.
Finally, whatever you choose to do, if you’re stuck, don’t hesitate to ask questions on the forums. Sometimes, it’s faster to do a Google search, but at other points, especially for fairly specific questions at a beginner-intermediate level, it can be faster to just ask.