Changing this fingerprinting info however does come with a price. It might cause the website to display pages in a different way, which will cause the site to look wrong or not to function properly. Lots of website developers rely on (at least some of) this fingerprint information in order to make the pages they display compatible with your browser. Hiding/spoofing this info will therefore affect this process and your browsing experience may be altered.
It's also possible that this fingerprinting is performed for legitimate security reasons. For example, a security-aware website might tie in the fingerprint info to the session or logged-in user, and will reject requests where the fingerprinting doesn't match (this would suggest to the server a request is now coming from a different browser). Here's one more example from another question where the website owner wants to test browser plugin versions of your browser (which is part of the fingerprinting data) before allowing access. I don't know of any specific sites that use this technique, but it is definitely feasible. Changing fingerprint data with every request will again cause unwanted behaviour.
I suppose that performing some minor randomized tweaks to specific headers will achieve a better degree of anonymity against fingerprinting without compromising the browsing experience. This requires some experimentation and will probably mean a compromise between security and usability.