Anything under a year, and you should make your builder handle it.
Having a carpet restretched by a professional carpet installer shouldn't cost much, call around and ask what it would cost. (You would have to take all the furniture out of the room first!)
This should be a warrantable condition on your new home, so go to the builder first.
As far as other options, I wouldn't cut the carpet either, and there really isn't another way to deal with it. What concerns me is that the screws are "backing out" in the first place. This indicates that the house was built with VERY green lumber, so I'm guessing there's a ton of drywall repairs you need to take care of as well. If the damage is widespread enough, and if your builder is going to fight you on taking care of it, you could hire a third party inspector (ASHI approved home inspector) to come out and get his opinion, which you can use in small claims court.
Regina might give you some good advice here too, but essentially, if the house is only a year or two old, you should talk to the builder.