Going out walking is one of my 'basic digger needs'. Alone most of the time - although usually with a dog now, but that hasn't been and isn't always the case. To be honest the dog I'm fostering at the moment is more likely to get me into trouble than out of it! ;)
I live on the edge of one of the UK National Parks - I'm not sure how that compares to the definition of a National Park in the States though (there aren't any ranger desks.) I have miles and miles of open moorland on my doorstep. There are patches of woodland too but it's not a dense forest type place. In other places I have lived previously there has been more woodland to walk in. I like walking in both.
I like walking alone and I am much more comfortable walking alone in 'lonely' places than I am walking through a busy town centre. As to how safe it is or not, I'm not sure. I think in terms of actual threats to my safety, people aren't going to come top of that list. A bigger danger would be having an accident in the middle of nowhere and no one knowing where I was, or getting dragged into a bog by a wayward dog! Sometimes I can be out for hours and not see another person the whole time.
I do still have some anxiety about other people as a threat, and this tends to ramp up when I do see someone else, even at a distance, but for the most part it sinks well into the background while I'm walking and I'm absorbed in other things.
I think, for me, I have decided that the benefits of being out walking far outweigh any potential risks and so I just say sod it and go!
I think if you are anxious about it, then, to begin with at least, maybe plan to go at busy times when there will be families about or try and find someone to join you and see how you feel.