Quotes below from Terez's interview database.

From a 1996 ACoS booksigning:

RJ: Access to the True Power is a matter of wanting it and the Dark One letting you. NOT black cords.

From the 2003 Budapest Q&A:

Q: What happens when Rand and Asmodean have this conflict and ...
Q1: ... and Rand severs his ties with the Dark One ...
Q2: ... and Rand severs some black ties. Isn't that ...

RJ: That was cutting off his protection from the taint and also cut off his ability, it was not like stilling them. It was cutting the ties that, most important to him, protected him from the taint on saidin, so he could draw saidin all he wanted to and never worry about the taint. But it was also those ties that represented his ability, or the conduits by which he could draw on the True Power. But it was not his ability to draw, it was not the same thing as stilling or severing, it was more like shielding.

From 2005 DragonCon:

Q: I'm trying to verify Rand's impression he has at the end of book four, regarding the connection he cuts from Asmodean to the Dark One. He seemed to believe it was the connection directly from that individual to the Dark One. So what I am wondering is, is that connection key to transmigration?

RJ: No, because what you are thinking of wasn't a connection. Rand only thought it was.

The 1996 quote and the 2005 quote are consistant, but how to factor in the 2003 quote?

As far as I can tell, RJ was trying to say, in that 2003 quote, that the cords represent the channeler's "conduit" by which he draws on a Power, either the OP or the TP. And, the Dark One uses that conduit to provide Taint-protection to the male Forsaken when they are channeling saidin.

So, the conduit is not a direct connection to the Dark One (as RJ said in '05), nor is it the mechanism that provides a person access to the TP (as RJ said in '96). But cutting the conduit not only severs the DO's Taint-protection, but also (temporarily) severs the channeler's ability to channel, either the OP or the TP.

Presumably, the channeler can re-establish the conduit himself, to be able to channel the OP again (since Asmodean was able to channel saidin again without anyone else re-establishing the conduit for him; I don't think Lanfear's partial shield did that), probably by simply reaching out to the True Source again. And, once the channeler did that, he could then channel the TP also, provided the DO granted access.

That's the best interpretation I can come up with, anyway. *shrugs*