We use a swivel for pulling fiber, but not for copper. From your description it sounds like the problem you are experiencing is the bucket o' string problem. In my experience, the bucket of string is great for pulling or shooting in, and then pulling wires. It totally sucks (IMHO) for pulling in with the wire. The strings natural coil/twist wraps around the wire great, just like it was designed to do that. I have almost completely quit using the bucket of string as a pullstring in place with the wire. We find many times that the second or third pull burns quite a bit of the insulation off the jacket or wire, the string is just too abrasive.
In larger conduits (2-4"

we often use 3/16" poly rope or 3/4" mule tape. The poly rope comes as a pull rope in fiber innerduct so we save the big pieces for pull ropes. The mule tape is a flat webbing product that comes on a big roll, has tons of pull strength, and lays flat pretty much when you pull it in (it is available in various widths and strengths, and often has foot marking). Personally I think the best way is to pull in your first run, pull it tight, push a fish tape with a ball on the end (I just make a wad of electrical tape) that helps the fish tape ride on top of the wires instead of intertwine with them as it goes in, then pull another run.
I suppose a swivel could help on the copper wires as well too, but I think you would need one swivel for each wire you pull, sounds like quite a bit of extra work. Also if you can lay the cables out before pulling it sometimes relaxes them, and of course, probably the best helper is good wire lube like Auqagel if you are pulling in conduit.
Good Luck!
It is only my opinion, based on my experience and education...I am always willing to learn, educate me!
Daron J. Wilson, RCDD
daron.wilson@lhmorris.com