I am afraid that the UX world depends on which your users are and therefore the true answer is that there are no solutions to your problem. There are only costs and benefits on choosing either one design or the other but this depends on the users. You should get in touch with them by doing ux research in order to get the proper insights that will guide your designs.
That being said one of the first things that come into mind is whether you have used the card-sorting technique because I see you have lots of categories and sub-categories. The card-sorting will help you see if you are on the right track with the proper trigger words that will help the users navigate more easily.
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You do not have to start from scratch. There are some well known patterns for navigation for mobile devices that could be a starting point on what you are trying to achieve.
For example, on your first screenshot, you are using the side-drawer which once again you have to map out the information architecture of the app and validate it with the users.
It is also not one way street what you are going to do with the content. For example you might need to get the users attention to the menu and one way to achieve that is to have the effect to push and shrink the content to the right. I will not include a screenshot of this because I cannot stress far enough that this is merely an example and nothing more.
If I understand correctly from your wireframes you are wasting most of the real-estate of the screen for navigation and you are sparing the content some small area at the bottom. My suggestion is to have the content fill the entire screen or try other variations and double check which is most comfortable with your users.
Another pattern that has emerged in cases where you have a two level hierarchy like yours (category and subcategory) is using a side drawer for the first level and a springboard/gallery for the secondary.
Springboard is the most common pattern, you have seen it in iOS and Android operating systems where the options(apps) are icons on the screen.
Gallery is similar but a peek of the content or a representative photograph can be used instead. Your content and your users will drive you to the better one.
The users have to do three taps this way. One to open the side drawer, one to select the category from the side drawer and a third to select the subcategory.
Let me know of your findings after the ux research. Hope I have helped.