What currency (there are lots of different dollars out there) and is that budget per person or for both of you?
I've routinely traveled in China for periods of this length, and regardless of where I am, have averaged about RMB 500-550 per day, all-in as a solo. Some days are more than that, some are less. That includes a mix of transportation (flight, standard and fast train, bus), private double room w/ensuite in a clean but simple Chinese hotel, food including some western meals here and there, local public transportation and a local day private car/driver once or twice, entry fees, and miscellaneous. I have a certain comfort level that I don't like to dip below--but many travelers on this forum can go lower. I don't skimp on entry fees--if I want to see it, I do it. My estimates are that if I were traveling as a couple this way, that average daily cost would go up by about 50%, to about RMB 800 average per day since you don't have to replicate costs for accommodation and car/driver.
Now that translates to about USD 117 per day for two people. So if your budget of $100 was in USD and for two people, you're not hopelessly far off, but you will have to either live more downmarket part of the time, or choose itineraries that don't take in every expensive national park in the book.
The biggest budget-saving tips I can generalize on are:
--In the cities like Beijing and Shanghai, you might want to consider hostels and guesthouses due to fairly expensive room rates in the more convenient locations. Many other places, you can get a decent private double room for RMB 150-200 per night.
--Consider taking sleepers in standard overnight trains which can save a lot of money on transport and accommodation--but at that travel period which is high season, you'll need to book in advance and soon.
--Generally city sights are still reasonably priced for entry fees, the budget busters are the natural scenic areas with high entry tickets + expensive add-ons like cable cars and shuttles. So the latter, you might want to keep under control. Every well-known village has a high entrance fee now, but pick just one or two and then go to unknown villages nearby without fees.
--Don't try to cover every corner of China in 3 weeks. The bigger the geographic spread, the more you'll spend on transport.
Since you'll be looping out of Shanghai, my initial thoughts are Shanghai region (including Suzhou, water town, Hangzhou)> Huangshan (big ticket item) and Yixian villages > Wuyuan villages > Luoyang > Xi'an > Pingyao > Beijing, > return to Shanghai. Another option might be Shanghai region > Guilin region > Zhangjiajie (big ticket item) > Qufu/Mt. Taishan > Beijing > return to Shanghai.
I like Yunnan and Sichuan but they can be exceptionally rainy this time of year, which can put a damper on things since most pursuits there are outdoors.
You need to figure this out fairly quickly so you can identify any critical transport links like trains that might sell out, and get them booked.