I have the 300 f/2.8 non VR lens (mine's the af-s II model, so I am not sure if you are working with the same one) and it is a great lens but one that does require practice and continued application of good technique (I could improve on the latter!). My suggestion actually differs from John and I'm sure he's more experienced than me, so take this for what it's worth...simply, what has worked for me...I would shoot aperture priority and set it to 2.8, then you'll have the highest possible shutter speed for the lighting conditions. If you find you're getting into bright sun, you can adjust as necessary, but if you are working with the same lens I use, it is best at wide open to f/4 (great everywhere else, but best wide open). Remember the 1/focal length rule, and for me, I generally add the digital 1.5 crop factor so 300 would be 1/~450 but I'm never comfortable hand holding until 1/800...that's me, though. Make sure to use the hood (flare will get you at 2.8), and maybe, just maybe, practice some handheld technique--feet about shoulder width apart, and I was taught to use what the person called a hunting stance (I've never hunted, so if this is not a true hunting stance, I wouldn't know) which he said meant position your body so that you are rotating your upper body about 45 degrees, don't lock your knees, trigger the shutter as you breath out, not on the intake, and of course "cup" the lens and let it sort of rest in your left hand, elbows in to the body and look for a tree or post to lean into...even with a monopod, I'd suggest the tree or post idea. Stay loose (which always works for me until I get excited and then my entire technique goes by the wayside!). I use my lens hand held in bright light when I'm in a hurry to catch a bird in flight and I don't have time to mount it and I find hand holding better than a monopod but of course, nothing beats a tripod, since you won't have a tripod with you, I'd opt for that tree post thing and practicing a bit with it hand held. You really may find hand holding a bit easier because of the flexibility. I have a gimbal (sorry, I know that's spelled wrong) style head, and I can't imagine not having flexibility with the lens, and I'm thinking the rigidity of the setup you're taking with you might be leading to that swaying feeling. I've also been told to use hand holding technique when using a monopod, especially that thing about not locking the knees, trigger the shutter when you breathe out, and cup the lens.
I don't know if it helped at all, but I can tell you that I was shocked that my hand held pics came out and my monopod pics by and large...not so good...so that's where I'm coming from. Best of luck to you!
