I'm no professional by far, but I'll give you my 2 cents anyway. Obviously your goal is to try to find out if there is a source for the hums and buzzes.
A lot of times we don't hear things in the room until they're recorded, so make sure absolutely everything is unplugged/turned off before you record. That is where the majority of all noises come from. I don't think that's what you're talking about though. You said you purchased a new mic (always state what kind of mic), and it helped out, so for the sake of ease & practicality, let's rule that out. Next step would be to bypass the mixer and plug directly into the 8-track recorder, making sure your connections are solid. There are three possible outcomes:
1. If you still get some humming, but less, it is most likely your cable. The obvious thing to do in that case would be to upgrade/replace your cable (cable is often overlooked but can make a HUGE difference). This is also one of the cheapest things you could take a "chance" on to reduce humming.
2. If you get no humming at all, it may be your mixer. It could still be your cable, though. First thing to do would to make absolutely sure your mixer settings and connections are correct & solid. Next, replace the cable. If that still doesn't fix it then take your mixer in to a professional.
3. If you get the same amount of humming, don't frat it could be the cable (yet again). It could also be your recorder, which I doubt.
I'm betting on the cable, but don't take my word for it. As long as you follow the age-old process of elimination you'll be sure to isolate the problem. Have fun.