The Zoom has mic preamps. The levels you're describing are good recording levels. That youtube guy with the video is giving pretty bad advice that just isn't true. You need sufficient headroom for things to sound clean.
I've never heard of "headroom distortion" before. The way he describes it is a contradiction. It just doesn't make any sense.
There are a lot of preamps out there that would probably be an upgrade from the stock ones in the Zoom. The thing is, is it worth it for you to spend the money? Everything in front of it has to be in order or you could end up spending a lot of money and not be able to notice a difference.
Meanwhile if you want better recordings you have to figure out if it's even the preamp that's holding you back. Having a really good monitoring setup in a room with some acoustic treatment is a big step towards being able to hear your mixes accurately. A nice preamp won't fix that for you.
In order of importance I'd say a solid monitoring chain is close to the top of the list. Headphones physically inhibit you form being able to hear things you need to in order to make a mix translate well to anything it might be reproduced on. After that, consider everything that goes into the source to make the recording.
Song, arrangement, performance. Has nothing to do with gear. Even modest gear doesn't do a whole lot to limit creative potential or proficiency. Can these areas be improved?
If you're recording instruments is there anything that can be done there? New guitar strings maybe? Better guitar? Amp? Intonation?
After that the mic is going to have a pretty big influence on the capture, but it's a guessing game until you can hear what you need to improve on. The monitors are essential. Mic placement is going to have a bigger effect on the sound than what mic you have. After you learn a few basics about polar patterns and microphone response, proximity effect and all that kind of stuff, you need to experiment. Moving the mics around changes the sound big time. Once you've exhausted the possibilities with the Zoom and a 57, it might be time for some upgrades in these areas.
That's how I'd approach it anyway.