I don’t have a lot of experience, but I will use what I do have to share a couple of thoughts.
There could be several things that are impacting how your sites are appearing on mobile, some of which might even be outside of your code (such as your phone). I could hypothesize a couple of reasons, but I think examples would help the community and I cite specific reasons why your websites aren’t showing as intended.
If your site looks fine when resizing it in a desktop browser or otherwise using dev tools to visualize them at smaller sizes, I could see how it would frustrate you if you went onto an actual device and saw something completely different. Again, examples would help us troubleshoot.
If it isn’t so much a technical issue with displaying your work and more of a design aspect, I find it can be easy to overestimate or underestimate proper size for buttons, text, etc. when you can’t see it on an actual device.I think with experience you will learn what are reasonable measurements for your different elements, but trial and error is always a good way to find what works. Try to imagine what you are making not on your giant desktop but on your comparatively minuscule phone, perhaps while you’re trying to get from one place to another in blinding daylight. Build your site to be usable in the worst case scenarios of accessibility, and you should be fine. This doesn’t necessarily have to be at the expense of intriguing design or eye-candy, especially since you can use media queries to optimize it for different screens, even if it means making it look arguably better on one screen for the sake of accessibility on another.
Regarding process, I can’t help a whole lot, but I’ve heard “mobile first” tossed around a lot, where you design for a small screen first and work your way up to desktop+ in comparison to building for desktop and trying to scale down. This might help, but your mileage may vary.
In short, examples will help us help you figure out how to best design for the wide variety of screens out there.
This is highly personal. I will operate on the assumption that many of your sites that you have made thus far were strictly for practicing development and not in use by others. If this is true, then you only need to update your sites if you want. If I were in your shoes, I would consider updating them for the sake of getting the experience doing so with material that you are personally interested in. Not to mention that you may be able (and probably should) use your work in your portfolio and have it look more impressive with responsive qualities done right. If you don’t want to invest more time into these sites, I don’t think it would matter if you went ahead and started projects from scratch with your new perspective and attention to responsive design.