Wait. When you uninstalled the application did you take note of it on your applications side? If you haven't done a fresh install where your application doesn't know that the app was removed by the user then your system will just think that it's a login.
When an application is installed you should always be getting a code which can be redeemed for the access token.
I'd need to see what your app installation flow is. Also if you are using cookies you might want to clear those too, they might contain information that is confusing your application. OAuth is not broken in Shopify, I've been able to install other OAuth apps.