I have an already published extension on Google Chrome Store and I already have an Oauth application running for it and I am currently paying for its assessment. I want to know if I can use the same Oauth app for the add-on as I have for the extension? Any help will be appreciated.
What you want to achieve is not possible as each application has its own permissions that need to be approved by the users.
Even if the extension has already been approved by a user, this doesn't mean that you can make the same user approve the add-on as well. Therefore, the authentication needs to be done separately for the two and the user has to approve consent the use for both of them.
Depending on the APIs you are using and what scopes you have set, you can take a look at the OAuth FAQ article here.
As for the add-on, it is important to test it before publishing it. Be mindful of the fact that the add-on will have to undergo app review before being published to the Marketplace.