Be careful with those emails asking to do anything to keep access to your account as they usually are scams / phishing attempts.
Bear in mind that passwords are not stored on the web browser cookies / cache but some web browsers like Chrome could save them in a special storage space. The cookies that could be deleted are those that save information about ...
Generally, Chrome keep signed in if you forget to sign out. But you can sign out from all the devices any time.
From Google Help:
Sign out from another computer
If you forgot to sign out of your email on another computer, you can remotely sign out of Gmail.
In the bottom right corner, click Details > Sign out all other web ...
Google does provide you the ability to have an admin account that doesn't have access to any of the Google applications other than the admin console and Groups for example. You will need to add a user with a Cloud Identity Free license and then assign that user admin rights.
You can read a bit more about it here and help improve that article as it is open ...