Hi Michael!
Xanth covered most of it, but I have some additional information that should help you with your problems, because those were the main problems I had to deal with at first too.
First of all, it doesn't matter whether you swallow or not, as long as it's a subconscious process. If you think about it you'll notice that most of the time when you swallow you don't pay attention to it at all, unless you're nervous. I advise to instead of trying not to have to swallow, just don't worry about it at all so you wont even think about it.
The second thing I'd like to point out, is that any bodily functions that may occur during your attempts don't matter at all as long as you pay no attention to them. I've never met anyone that doesn't move around, or cough or sneeze or something while they're asleep. I've been told I do stuff like this while I'm asleep as well. Some people also talk in their sleep or get up and do something (while still completely asleep.) The point I'm trying to make with this is that your consciousness pays no attention or cause any of these actions, which is all you need to worry about silencing. Try to let your subconscious mind deal with all the physical body stuff, while you and your consciousness become one with your spirit.
That being said, it doesn't really matter if your eyes or open or not. It's just easier for most people to concentrate when their eyes are closed because of the darkness. So if possible get yourself in pitch black darkness so even if your eyes were open, you wouldn't be able to tell a difference.
There are some methods that require your eyes to be closed though. Even some that require your eyes to be open. I've had OBEs with my eyes open before so it works, as long as it doesn't cause a distraction.
So good luck if you have any questions feel free to ask!
Xanth covered most of it, but I have some additional information that should help you with your problems, because those were the main problems I had to deal with at first too.
First of all, it doesn't matter whether you swallow or not, as long as it's a subconscious process. If you think about it you'll notice that most of the time when you swallow you don't pay attention to it at all, unless you're nervous. I advise to instead of trying not to have to swallow, just don't worry about it at all so you wont even think about it.
The second thing I'd like to point out, is that any bodily functions that may occur during your attempts don't matter at all as long as you pay no attention to them. I've never met anyone that doesn't move around, or cough or sneeze or something while they're asleep. I've been told I do stuff like this while I'm asleep as well. Some people also talk in their sleep or get up and do something (while still completely asleep.) The point I'm trying to make with this is that your consciousness pays no attention or cause any of these actions, which is all you need to worry about silencing. Try to let your subconscious mind deal with all the physical body stuff, while you and your consciousness become one with your spirit.
That being said, it doesn't really matter if your eyes or open or not. It's just easier for most people to concentrate when their eyes are closed because of the darkness. So if possible get yourself in pitch black darkness so even if your eyes were open, you wouldn't be able to tell a difference.
There are some methods that require your eyes to be closed though. Even some that require your eyes to be open. I've had OBEs with my eyes open before so it works, as long as it doesn't cause a distraction.
So good luck if you have any questions feel free to ask!