Well, I'd suggest speaking to your sensei because there are a lot of variations in how cuts are "supposed" to be done. So grain of salt to anything I might tell you since it may conflict with the advice you *should* use -- your sensei's. Anyway, a heavier sword, in my experience, will help people cut straighter because it simply has more mass and hence inertia. Get it started and it will go where it wants. Most bokken are quite light, lighter even than a real sword for that matter. And the balance is different. Which is one reason I have a bag full of bokken of all sorts of different shapes, sizes, woods, etc. Of course I also have a few shinken to play with, so I get a lot of practice.
Side benefit of the job. But... to your point. If the lighter ones are problematic that usually points to something in your body you're doing incorrectly. Check your grip. Check your hips to shoulders to arms positioning. Some styles square up others go more oblique. Fix whatever needs to be fixed there. It can also help to focus out far in front of you and think of extending a bit more -- getting the tip out and just letting it drop once it's out there. But if you're slightly skewed, rotated, leaning, whatever that will translate into weirdness in the arc. Watch yourself in a mirror if you can. Then practice a whole lot and slow down. There is absolutely no value in making a lot of bad cuts. Only push up intensity once you get into a groove of good cuts.
I find that due to a series of weird injuries and a recurring myopathy (which causes me to change form which helps me get weird injuries) I am constantly fixing my form. My body doesn't cooperate with me some times so it takes a lot of focus for me to prevent bad habits. So I'm always working with myself, negotiating, trying to figure out how to fix things that creep in.