How to manage cookies in a Chrome extension?

Question

Google Chrome

Hello everyone, help with a problem:
A user downloads and installs an extension for the Chrome browser. This extension works with web requests, simulating user activity on multiple sites via the user's IP.
The problem is that when working with cookies, they are stored in the browser. If the user enters the site through the same browser, then there will be those cookies with which the extension worked (and some sites transmit sessions in cookies and a lot depends on them)
Example: I went to the test.com site, logged in, and it passed the session to me in cookies. I installed a browser extension that automates the work on this site but from a different account, the extension starts working, respectively, it authorizes on the test.com website under a different account, and this automatically ends the previous session in this browser, but it is necessary that the user does not notice this.
If nothing is clear above, which is most likely, I will try to ask more clearly.
How to make the Chrome browser extension in the background create the "Incognito mode" of the browser and work in it, but so that no new browser window pops up.
Or how to implement work with cookies in the extension so that cookies do not interfere with the user's work in the browser?