The fact that the mobile app can still access LinkedIn means that there is an authentication back door that it uses. What does that back door look like? Clearly it does not care about your password.

If someone studies the mobile app and figures out how it does this authentication trick, then we'll know the damage. Is there a universal authentication where you just need to know the person's name to connect? Is there a table inside of LinkedIn that can simply grant access? (If the blackhats can get a dump of the user table, they probably can manage to insert things into other tables.) Whatever the mechanism turns out to be, it provides a way for the blackhats to bypass security and access your stuff even if they do not know your password. And the odds are depressingly high that they'll be able to do it even if they never knew your password.