What, exactly, do you want the tour to do, if you're handling your own transport and accommodation?

If you do a search here you will find several trip reports from people who have traveled independently in China. You will also find several threads discussing the up vs downsides of tours in China.

There is no shortage of taxis, or of travel agencies, in China and you can arrange a day tour very easily when you are there if you feel you need one.

I agree with Thursdays. Go ahead and make your own air and hotel reservations and go. If you find you want a day tour, you can arrange it easily on the spot, but you can also do it on your own. Do wait until you get there to arrange any tours, as I expect you will find it much easier nad less intimidating than what you are expecting.

It will also be considerably cheaper arranging it on the spot (as hotels and internal flights would be). Don't forget to bargain, and don't tip.

A lot of hotels/hostels also arrange their own tours. Once your arrive at the hotel then you can decide which ones you want to join.

contact a china agency and see if you could join a tour for westerners...

or

contact a usa agency and see if they would sell you just the land portion without hotels

Bob's suggestions mean you are stuck with an expensive tour group for your whole trip. If you wait until you are there are book day trips as you wish, you can pick and choose as you wish.

Independent travel in China is easy. You can easily book your flights and hotels on the internet. Where do you want to go? Once you are there, there will be many, many drivers who will take you wherever you want to go for the day. I assume you don't speak Chinese- you can get a translation app and use that to show drivers what you want to do, and always have your destination in Chinese characters and you should be fine.

The only issue is foreigners can't book high speed train tickets in advance, but unless you are traveling on a major holiday (which you shouldn't do anyway, the crowds are absurd), your hotel can book the tickets for you.

As it was just said, booking on the spot is even cheaper, but depending on the standard of hotel you want, this might not be an option for you. Hostels and below 4* hotels you can book on the spot. Personally, I prefer piece of mind from having something set up beforehand, but to each their own.

We are traveling independently in china and are now in Beijing. I can attest to the fact that traveling here is easy. You can book day tours once you arrive, if you want them. If you have a good guidebook, you can see sites without a guide. Often there are gyides to hire at the main sites ot audio guides if you want them. The taxis are very cheap and the subway system is easy to maneuver so getting around is no problem.

Coincidentally, we were visiting the summer palace when we ran into relatives who were on a guided tour. We tagged along for approx 1 1/2 hrs with their group while they raced through the grounds, stopping only occasionally for toilet breaks. When they departed we returned to the beginning to see the areas of the grounds that were missed. I have never been so happy to be traveling independently! The group tour missed the nicest part of the grounds (likely because there was a slight surcharge). For your info, our relatives are very happy with their tour, but one never knows what can be seen by wandering independently.

Seriously, you will have no trouble traveling in China alone. We had a little book "Me No Speak Chinese". We only had to use it once in a month. Mostly everywhere we went we met English speakers and in the cities, quite a few signs are in English.