Generally there is no need to book ahead. July 4th is a national holiday - Independence Day - almost everyone has that day off as a holiday and people tend to drive around and camp. If you think you are going to need a hotel or be in a popular vacation spot (like Yellowstone National Park, for example) on/around July 4th, then you might need to book something in advance.
In the west you can camp on forest service or BLM (Bureau of Land Management) land for free. How do you know if you're on that kind of land? Well, I don't know, hopefully someone can help out with that, but the Adventure Cycling maps have a lot of information about where to camp, including free city parks.
National Parks (Yellowstone) may have "hiker/biker" spots reserved, and on the coast most state parks have these too. You can NOT camp outside of campgrounds in national or state parks. Most forest service campgrounds are pretty relaxed, so if you get there and they are full, the camp host will often find somewhere to squeeze you in. You can also camp at "RV Parks" = campgrounds meant for huge motor homes - they are usually not as pleasant but have good facilities like hot showers, laundry and often a pool.
If you do get to a campground that is full, you an often find someone willing to share a site - I've only had to try this a few times, but it has never failed me.
Have fun, you picked a great part of the country to ride.
On a side note, the Denver Airport is far away from the city, if you don't want to ride out of the airport there is a local bus SkyRide http://www.rtd-denver.com/skyRide_SubHome.shtml
that can take you into Denver or to Boulder if you are going North, for a very small amount of money - and you can put your boxed bike on the bus.
You didn't ask but I have a suggestion about getting to the Trans Am route. Even though it will be very hard because it is a huge climb and you will be at a high altitude, may I suggest that you head north to Boulder, then Estes Park then Trail Ridge Road through Rocky Mountain National Park. You can connect to the TransAm in Kremmeling, CO. You'll want to check the weather forecast, but if you have nice weather it is really spectacular.
I live in the area, so if you haven't sorted out your connecting route, please fee free to ask here if you want some suggestions. There are a lot of ways to connect, a few of the ways you see on a map look good but are not, but there are lots of good choices.