The Lumia 950/XL are known for oversaturating colors of the photos taken with them. Some may like it, some may not. Alas, there is no known way to disable the 'applying finishing touches'.
You may have known this, but I'm going to say it anyway (and I might be wrong about this as well). Shooting RAW/DNG files will create two copies of your photo, one in the oversaturated JPEG that you're complaining about, and another RAW/DNG file that is untouched by the 'apply finishing touches' process (according to others from this forum. I am personally not sure about this). You'll get to edit the photo's RAW file using Photoshop and whatnot to 'process' the photo to your liking. Basically, instead of the photo being automatically processed by the phone a.k.a. "applying finishing touches", YOU are the one who manually adjusts the final outcome of your photo.
And to answer your question of "Do you have to shoot in RAW mode? Or what?", no. Shooting RAW will still get you the oversaturated photo, but it also gives you the chance to edit/process the photo yourself via the photo's RAW file.
There is a way to help you through this, though it may cost you a bit (and no, it's not buying yourself a new phone). There is an alternative camera app in the Windows Store, called ProShot, costs about $4 if I'm not mistaken. The app's developer claims that the app uses it's own API to process the image that you captured, so that the final photo is different from the ones you'd get from the default camera app. Some users have supported this, but I personally see little to no difference between the photos I took with the default camera app, and this ProShot app. Maybe my eyes aren't trained to see the difference, so your mileage may vary.
A word of warning though. The app has been updated recently and a few users complained that the app is buggy. This is probably due to the recent updates to the OS, which causes the app to be incompatible or whatnot. But then again, these users didn't mention what phones they were using the app on, and personally, the app worked well for my 950 last time (Anniversary Update). I would download it again on my 950 to see how it performs now, but I'm quite busy at the moment.
Also, you have to make sure that you are using the Windows 10 Mobile version of the app. There are 2 versions in the store, one is the older version that was made for Windows Phone 8.x. Both of them are from the same developer, and both are paid apps, but if you bought the wrong one, you'll still have to pay for the other. Just read the descriptions of the app and hardware/software requirements before buying, if you plan to try the app.
Oh, I just remembered that they do have free trial, but it's only for a few days, and the trial version has very limited functions compared to the full, paid version.
Also worthy of note is that, if you somehow like the app, the last time I use it, I cannot set it to be the default camera app. Meaning that I cannot adjust the phone's setting so that it will launch ProShot instead of the default camera app when I launch the camera via the camera shutter button. This was the main reason as to why I ditched the app.
Hope it helps. Cheers!