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If you know the login details for the browser, where your passwords are saved, you can just login to the browser on a new computer and carry on as before. Why your current company would allow you to do this I am not sure, seems a little slack to me.

You are always logged on? or always automatically logged on? If it is the latter, does your current employee have some single sign-on software? If so you should be able to access this and get the passwords, before you leave.