Daily Mix is intended to work this way. It's ought to give you songs that you already know and like, not the new ones.

I'm finding that even discover weekly is giving me songs that I've already listened to a lot. I wish there was something I could do to tell it to stop that.

Maybe there just aren't many songs out there that I truly like, but every month or so I do still stumble on a couple that are truly awesome. Does anyone have any suggestions on a similar service that could suggest more songs to me in a better way?

Yeah, indeed, sometimes they seem to shove some bands down the throats. At least in the radio mode you can use the "thumb down", and I hope they're checking which songs I keep skipping. But, for the best results: try listening to other people Discover Weekly playlists. I got few folks on my list with a similar taste, so I won't hear anything offending on their Weeklies, yet the lists are quite often totally different.

interesting, but then the challenge becomes finding people who have similar tastes to me :\