I used to have the same problem and would not sleep well b/c of how tense I was and would wake up and listen to everything. I don't know what made me change but I now feel very comfortable by myself in the woods. I do have a few theories though...
First is just the pure odds of what there is to be scared of. On the east coast in reality there is just not that much to be scared of in terms of wildlife. Black bears are not viscous animals and if you take the proper precautions like hanging your food then you should be fine. Also sit down and listen to how much noise a squirrel can make, something small can sound much much bigger.
Second is finding a camp site I am comfortable with. I don't mind camping around other campers as most every person I have ever met on the trail is friendly and harmless. I do prefer to be off the trail some and in an inconspicuous spot. For me it is just peace of mind.
Third is taking solace in the fact that I have never known or heard of anyone I know that was hurt in the woods or met anyone or any animal that had any malicious effect on a hiker, and I believe most stories you read of a hiker and a bad animal encounter was probably a little inexperienced in the woods. What does worry me is lightning but that is another story.
My first step to getting over my constant waking up to every branch that snapped or leaf that rustled was ear plugs. They helped me stay asleep and I just learned not to worry about what I couldn't hear. After you learn that what you were worried about has never led to anything other than lost sleep it becomes much easier to relax.
The other thing that really helped me was reading some of John Muir's writing. Knowing the amount of time that he spent in the woods alone and how well he came out of it all is somewhat comforting. I honestly believe a positive attitude and calmness with where you are and the situation you put yourself in will go a long way.
I would also lie to note that I do not carry a gun although I am a gun owner and I do not believe that having one with me would make me feel safe.