During our 16 years of full-timing and traveling constantly, we rarely made reservations. It's too big of a commitment to be somewhere at a specific time. If we don't like a place or area we'll leave earlier than plan and if we really like it we'll stay longer. So..... guess what my recommendation is?
A Sprinter would be an ideal RV for you. It has plenty of room for 1 or 2 people (We've even met full-timers with one) You seem to be very active and wouldn't spend lots of time inside anyway.
Many national parks have campgrounds that don't even accept reservations. Those are the ones we'd head. However, you need to get at the campground early morning when folks are leaving. Park nearby the previous night. With your small RV you'll have many choices of sites. A few times we just went directly to a reservation-type campground in a national park, stopped and secured a cancelled site.
Each national park has an awesome web site giving lots of good information such as campgrounds (this is where you'd check if the campground accepts reservations), things to do, closings road issues, special alerts, etc. Check them out.
For instance, Yellowstone has 7 campgrounds inside the park that don't accept reservations. They would be perfect for you because some are geared for smaller RVs and tenters. Most of these kinds of campgrounds are in the best areas of the parks and very scenic. Next door is Grand Teton Nat'l Park which has two huge (300-site) campgrounds that don't accept reservations. They rarely fill. We like Gros Ventre. Grand Teton has a wonderful paved biking trail. National parks won't allow off-trail biking unless designated for it and usually don't allow biking on any trail unless it's strictly a bike trail.
Yosemite is not a favorite park of ours because of the awful traffic and masses of people. I can't see riding a bike in that park.
Keep in mind that your Sprinter doesn't need to be hooked up to electric all the time so you have more options in finding sites. We actually prefer dry camping spots as they are usually more scenic.
Outside of the national parks are many, many national forests campgrounds which we love. Some are on streams and lakes and all in places that would be good for biking. You could do day trips into the national park for hiking, climbing, siteseeing and plan to do your biking in the national forests.
Here's a great site for finding national forest campgrounds with good directions and explanations of the campsites and roads getting there. You first have to know the name of the national forest and you can get that from any map.
Coastal California is busy and all over the country any park that is close to a major city or popular area is busy... such as Rocky Mtn. Nat'l Park. Even coastal has some national forests nearby so look into those. State parks tend to fill but there are always cancellations but if you can find some that don't accept reservations give them a try by coming in early morning.
You might also check city and county parks as you travel for your base and bike from there. We found some very nice ones. Moab, Utah is awesome for your kind of biking and there are lots of BLM campgrounds.
Personally, I think with your 3 months of time that you shouldn't go all the way to Virginia or the east. Concentrate in one area and it will be more relaxing for you.
Also, you don't have to stay at WalMarts. Here's a site that gives free or low-cost camping.
When you enter a new state stop at a Visitor Center for camping information, especially on city, county or other types of places. We've stayed at some gorgeous fishing access sites on rivers and lakes and some states have a booklet of where you can stay at these places. Montana, for one. You don't have to fish to stay there. It would just be a nice pleasant place to stay a night or two.
National Wildlife Refuges sometimes also allow camping... usually no hookups but oh, the quietness and scenery and great biking.