First, welcome to the ACA Forums. It sounds like you have a great trip planned.
Based on your post, I am assuming you are doing the TransAm (TA) Route or something very similar. If you are not doing the TA, I would strongly suggest you consider that or another ACA route until you get a month or two of touring in the USA so you are comfortable with going independent. Yes, 3 months is a very reasonable time. You very well may start out slow, i.e. 30 miles per day, but within a few weeks your body will adjust, even if you are overweight and/or not as fit as you could be. I have met dozens and dozens of 60+ overweight riders over the years who all basically say the same thing, the first few weeks are a bit rough as the body adjusts but then it gets good. Do not be surprised if you lose weight AFTER you body adjusts. You may even gain weight initially as your body freaks out to this new routine and thinks it needs more calories. I almost always gain a few pounds on tours less than 2 weeks long and lose weight on tours over 5 weeks long.
I would guess about 40% of the riders on the TA (or most long tours) are solo riders. The TA is a very good route and you will most likely not have problems with animals or humans. Do not believe all you read about in the news. Americans are just like everyone else in the world in that they want to raise their family, go to work/school, and live their lives without too much interference. Unfortunately, that does not sell newspapers so they have to sell the stuff that is "news worthy". Assuming you do not go looking for trouble, you will be fine.
As far as traffic goes, the USA is a fairly large country so drivers are different in various parts of the country but generally speaking, the drivers on the TA are overall OK.
As far as body soreness, without seeing you riding and your body position, it is difficult to give a proper reply. That said, it is normal for your body to have sores as you described until your body "toughens up" which can take a week or so of daily riding. If you are in a "racing position" you could aggravate the symptoms you described so you might try a more upright position as that reduces the force on the arms and wrists and straightens the neck some.
Tailwinds, John