As a guy I can't really provide any direct advice, other than I think that state parks are some of the safest places you could camp.
I met a gal last summer (in a state park), she started her tour 100,000 km and seven years before I met her. Started in Africa, then Europe, then Asia and Japan, Australia, back to Asia for a while, then I met her on the Pacific Coast in Oregon. She had been traveling all this time alone. The first photo was of her packing up her bike in the morning to get going. Her bike frame was covered with the stickers of the various countries and cities she had visited. I saw her again later that day, took another photo of her.
If you need a little more experience bike touring to feel comfortable traveling alone, take some self supported trips with Adventure Cycling to get used to traveling by bike. On those trips it is common for people to travel alone during the day or in small groups because everybody has their own pace that they are most comfortable with.
Someone above mentioned a can of pepper spray. If you have something like that, just having it may make you feel safer. Or a bigger size like used for bears, but that would not be very convenient to carry around with you in a park.