This is probably stupid, but with the sub unplugged from the receiver, but still having the cable plugged in and picking up the radio with a shielded cable, are you sure the connection on the sub is grounded for shielding?
Tried wrapping aluminum foil around the speaker for crappy makeshift shielding? (my probable stupid idea)
Although I have never experienced this precise problem, I have had humming issues with subs in the past and have been intrigued by your issue so I did a bit of reading up. What follows is nothing more than some of what I've read so take it with a grain of salt (I'm definitely not an EE or anything).
I think what he means is that even the most heavily shielded cable can potentially still act like an antenna unless the shielding in the cable is connected to a ground. It sounds like your subwoofer interconnect cable shielding might not be working properly. This could be because the radio transmitter is really really close (in which case you might be SOL). But it also could be because the shielding in the cables you're using hasn't actually been connected to a ground. This can happen for a number of reasons.
1. The shielding connects to ground through connections at the cable plugs. Sometimes these connections fail. And in some cases, a device you're connecting to isn't designed to ground the shielding (sometimes the case with subs). You can test this out by using temporary replacements if you have access to a spare sub and receiver. You will want to switch out your sub for a replacement and then your receiver. If your sub isn't designed to ground the shielding, you need to rely on the receiver's ground. If the receiver isn't properly grounding the shielding (either because you've disconnected the sub cable or because of a fault with the receiver or because the receiver's electrical outlet isn't grounded), you might have an effectively unshielded cable.
2. Some subwoofer cables are made to be connected in a specific direction. This isn't because of some black magic audiophile stuff. This is because the shielding on some cables is connected to the plug at one end only (i.e. the cable is designed to reach ground through one end only). Cable companies sometimes make sub cables in this manner to try to reduce ground loop problems (humming). Typically, such cables will have an arrow at one of the cable. The arrows show the direction the signal should be travelling (i.e. from the receiver to the sub). The end of the cable to which the arrow is NOT pointing is where the shielding is connected to the plug (i.e. the shielding reaches ground through the AVR). If you've been using such cables and connecting them in the wrong direction, the cable shielding might not be grounded. I doubt this is the reason if only because I own three different monoprice sub cables and none of them are unidirectional. But I thought I would mention it if only because switching the direction of the cables is easy enough to do.
3. The tin foil idea might work if you can find a way to ground the foil. I suspect it would work better if you wrapped it around the cable (and grounded it).
Good luck. These problems suck.