There aren't a lot of great apps out there that don't require iOS 8 or higher. A lot of really useful features were added, and all the best app developers left older iOS versions behind ot support them. The best Notes apps are all going to offer extentions that let you save notes without having to leave the browser or app you are already in. You can't do that pre-iOS 8. Pre iOS 9, I don't even recommend the built in Notes app.
Anyway, if that is what you're stuck with, then your workflow will be to copy lyrics in one app (Safari probably), then open the notes app to paste them. In that case organizaton and searchability is probably going to triumph any other consideration. FormConnect might be worth looking at. It's a form based database tool.
Plusses: You can create a form with fields like Author, Date, Artist, or Genre and organize/search using them.
Minuses: You'll need to create your own custom form. Not difficult, but some learning involved. Eventually if/when you want to move to another app exporting won't be as simple as moving a bunch of text files.
Given the limitation, you might also want to look at some text editing apps, not just notes apps. Many of these have good organization tools, and powerful searches. The only one I have right now that is iOS 7 compatible is Daedalus Touch; an interesting app that take a different approache to text documents. Since it will save documents to several cloud services (like DropBox), it's quite flexible; but it's organization method may not be what you are looking for.
I'm afraid that's all I can offer.