HI Scott,
First, I just want to say that I completely understand how confusing and muddy this is from a change management perspective. It is a lot to manage and it isn't always crystal clear when versions change and such. Hopefully I can clarify this and help you get what you need to be successful.
I checked and the latest is 2019.2 currently. You are correct that the authorizations numbers change after 10.5, but that will still be true whether you upgrade License Manager or not. What that means is that these new licenses need to be loaded into the ArcGIS License Manager in order to be made available to the clients you are authorizing. If you want, you can simply upgrade the License Manager to 10.6 and then wait for the next upgrade to make the leap to 2019.2.
You can update your License Manager at the same time you upgrade your ArcGIS Enterprise or ArcGIS Desktop software. Alternatively, you could update on a yearly basis. Its really up to you but ArcGIS License Manager should be the first in line to get updated.
The update may sound like an upheaval but unless you are changing the machine where it is installed, it should be relatively easy. If you chose to make the jump from 10.4.1 to 10.6, just load the installer, stop the licensing service, install 10.6 (it will detect and replace the 10.4.1 install), and then restart the license service and you are good to go. The same license you had before will be there but you will have the ability to add higher level licenses up to 10.6.
This documentation shows what that looks like. Even if you do not have live support at the time of the upgrade, I would still advocate for logging a support ticket if you need it. We want you to be successful and it always helps to have the resources available if needed.