They have this article where you have to enable users to start a trial, then try to login to Teams and start a trial then you should be able to assign trial licenses to users. I'm not 100% sure this is live yet or not since I don't have a tenant that doesn't have a Teams sku! But worth a shot! https://docs.microsoft.com/en-gb/MicrosoftTeams/iw-trial-teams
Yes you cannot use free teams as an organization. It’s only for free Microsoft accounts. They do offer a trial for 1 year however but after that time you will have to purchase licenses to continue use. I would recommend contacting support. Since you do have office 365 you should be able to get support for getting s trial going.