I did try to rewire the Sherwood and I am getting the same hum. Let me try to jiggle the white wire and see.
Side note: Do any of you do skype? I feel like with your help, in ten minutes this could be fixed. I would be willing to throw a cup of coffee or a beer your way

If not, I can just keep using the forums.
Not at home now, but can link up via Skype later today when I’m back home. But hums can be somewhat subtle to track down, but in the context of a TT, it’s usually a grounding issue.
If the ground wire is securely attached to the receiver, I would verify your headshell leads are firmly attached to the cartridge. Typically, green is the ground for the red/right channel and blue is the ground for the left/white channel.
If you’re only getting hum from one channel and not the other (you had said the white RCA gives you hum), I would check the white and blue head shell leads to make sure they’re securely attached to the headshell and then jiggle the white RCA attached to the receiver (you can giggle it both at the RCA attachment to the receiver as well as where it enters the TT). But while you’re doing that, might as well double check the red/green connections and the red RCA too.
Sometimes the cable itself gets frayed internally where the connections are marginal and jiggling will make it better or worse.
Do the RCA cables enter the TT body such that you can’t see where it goes internal to the TT? I ask, because sometimes TT’s have RCA jacks that allow you to connect your own RCA cables. Sometimes these jacks are internal to the TT, sometimes they’re on the back so you can see them. If there are jacks, you could try another set of cables or at least make sure they’re securely attached there.
If they go inside and you can’t see them, and you’ve tried everything else mentioned above, an option would be to try and open the TT and see where the RCA cables go and if anything looks loose. But how easy this is varies from TT to TT, so I wouldn’t go there unless you’ve tried everything else.
But the fact that you get hum on both Teac and Sherwood points to an issue on the TT itself.