The Great Ratline Problem has frustrated modelers literally for centuries, but it really has two solutions: practice and patience. The more of the former you have, the less you need of the latter. The problem doesn't need to be as intimidating as it seems to be for so many people. Once you get the hang of it, rigging ratlines isn't any more difficult than lots of other jobs in building models.
I'm a longtime believer in two methods of rigging ratlines: the "needle-through-the-shrouds" method and the scale method, with clove hitches.
If you're having trouble keeping the ratlines horizontal with method #1, I can suggest two possible problems. One - there's something odd about the line you used for the shrouds. That seems unlikely; I've used quite a few different kinds of line for the purpose, and never had that particular problem. Two - you need a little more help getting the ratlines lined up, and keeping them that way. My favorite technique is to rule a series of lines, about a scale foot apart, on a sheet of stiff white paper (file card stock or something similar). Cut a piece of this ruled paper so it fits between the channel and the masthead, and shove it into position just inside the shrouds. Now you've got a guide to where each of the ratlines ought to be. As you rig them, put a drop of white glue on each end of each ratline, where it meets the inside of the foremost and aftermost shrouds. That should take care of the problem.
It sounds like that's just about the way you've been doing it. If so, the best suggestion I can offer is - keep trying. That system does work. One big reason so many people have trouble with this sort of thing is, I'm convinced, that they give up to soon. The muscles in the average adult modeler's hand are perfectly capable of doing this job, but they aren't used to it. They need a little training - both to go through the necessary motions and to work out the necessary coordination with the eyeball. I don't spend much time on model building these days; years sometimes go by between one rigging exercise and the next. I'm always surprised at the way my fingers forget how to do things - but they always pick up the tricks again with a little practice.
I use the same sort of alignment aid for ratlines that are clove-hitched. There's no denying that it does take a little practice to get the tension right. One little tip that may help. Start at the bottom. If you find that a ratline is too taut (i.e., that it's pinching the shrouds together), slide it upward a little so it becomes the next ratline up the mast, where the shrouds are closer together. Then try the one underneath again. Nobody will know.
As in so many other aspects of sailing ship model building, there's a short but rather steep learning curve here. My guess is that it may take you fifteen or twenty minutes to rig the first ratline on a given mast. But by the tenth one you'll have cut that time by at least 50 percent, and by the time you reach the top you'll be rigging one ratline per minute.
My personal taste leans toward smaller scales, and there, as Millard suggested, wire does come in handy for ratlines. I suspect Forum members are tired of seeing pictures of this model, but for the benefit of any newcomers who aren't already asleep here are some shots of my Continental frigate Hancock, on 1/128 scale: http://www.gallery.drydockmodels.com/hancock . The shrouds and most of the other rigging on this model are silk thread, spun up on an incredibly crude "rope-making machine." The ratlines are nickel-chromium wire, from a spool that a friend found years ago in a military surplus store. It's about .002" in diameter, and seems to be a nearly ideal material for that particular purpose: stiff enough to hold the "sag" between the shrouds, but flexible enough to tie in a clove hitch. Unfortunately I have no idea where to buy such stuff.
Hope that helps a little. Good luck.