Mr. Smith, I have a NH 851 AutoWrap chain baler that has/had the same problems you describe. Here's what you might look at:
There are two or three adjustment on each side of the baler that will "tighten" the bale up a little. On my baler there is a cable that runs over a cam or sheave at the very top of the baler. The cam (or sheave - what ever you call it) should be up, that is the cable runs over the top of the cam. This puts more tension on the chain and bars.
The second is the bale tension spring located on the sides of the baler. Mine has a cover over it so it's not visible until you remove the cover. The spring controls the bale tightness. More spring tension the tighter the bale. Mr. Smith, BE CAREFUL WITH THIS SPRING. As the other gent wrote, get a manual for the baler to learn how to work on it safely. There will be a "fine" tension adjust which is a screw (3/4" all thread rod) with a nut on it that can be tightened, and a "course" adjustment which is the starting point for the all thread rod . There are holes and pins which, when the spring tension has been released, you can remove the pins and re-pin in a different hole. Of course all of this depends a great deal on what you are baling. The coarser material does pretty good. The finer (Bohia, for example) doesn't bale very well no matter how tight you have the tension set, IMO.
The string. When I talked about this issue before on here some folks po-po'd my comments but I'm telling you that cheap, orange string I used to buy at TSC won't work. I've had the cutters sharp enough to shave with and they will not cut that string. I use the blue and white or red and white string and it work just fine. The differences between the "good" string and the TSC (I'm pickin' on TSC - the cheap stuff is available from several sources) is obvious when you put the two side by side. The string is trapped between the knife (sickle blade) and a knife backing plate. The cheap string simply bends or snakes around the knife while the better string is much stiffer and doesn't bend as easily. So when it passes over the knife it will be cut. Solved my problem, hope it works for you. One last thing, the knife should be a smooth sickle blade, not the serrated type. They're a little harder to find in my part of the country but that's what you need to use.
On my baler there is a panel at he back bottom of the baler (door). This panel controls the "fines". If you want to bale the "fines" the panel should be on (in place). If you want to let the "fines" pass through the baler then remove the panel. Again, it depends a lot on what you're baling. Bohia (I hate that stuff as hay) when it's dry and ready to bale is almost all "fines" so without the panel in place it will all pass through the baler and not get wrapped up. Even when the panel is on there is a lot of material that doesn't get wrapped. There' a lot to be said for the belt balers. But when all you have is an old chain baler then you have to make do.