Migrate Logged-in users from org auth server token to custom auth server token

The requirement is to validate the tokens locally in our .net code but can not achieve that as currently as my client’s production tenant does not have the oauth2/detault i.e. using org auth server. So if we plan to migrate the applicant to custom default auth server, then there is an issue with existing logged-in users how we can keep them logged in.

For time being we can validate the existing users from okta as tokens are short-lived so we do not need to support this for a long time, but the token can be refreshed by the refresh token.

My question is that, is there a way to provide users new token from custom default auth server when they request for a new token from refresh token without asking them the username and password again for the token(i.e. without logging them out)?

Or is there any other way to migrate existing logged-in users from org auth server token to custom auth server token?