I have a Sitecore 8.2 project that I'm upgrading to 9.1.1. I installed a fresh instance of Sitecore 9.1.1 and moved my old project into it. I have it pointing to the old Core, Master and Web databases, and everything seems to be running good. My question is do I need to upgrade my databases at all?
I don't think I need to install the "upgrade package" unless it includes any database upgrades.
Yes there are changes to the database. I have performed this upgrade in the same way you want, and you need to run the Database Upgrade Script found under Configuration files for upgrade at: https://dev.sitecore.net/Downloads/Sitecore_Experience_Platform/91/Sitecore_Experience_Platform_91_Update1.aspx