For some reason, and it has never done this before, when I launch Skyrim with SKSE from either Nexus Mod Manager or from the actual launcher .exe, it launches the Skyrim launcher from Steam, instead of just starting the game, and because of this, now a lot of my mods don't work, as they require SKSE. Help?

Witht he skse launcher i'm guessing you use a shortcut? right click and in properties see if it's still pointing to the actual skselauncher.exe

No, I don't. I typically launch Skyrim with SKSE through NMM. But when I tried that, it launched the Steam launcher. So I tried directly launching it through the SKSE Launcher in the Skyrim directory. But it did the same thing.