In order to change your ID (whether you are part of the preview program or once the feature officially rolls out), you can go to the Settings menu or complete the change on the Profile page of your PS4. The testing phase is slated to wrap up in November.
While all PlayStation 4 games published after April 1, 2018 will be compatible with this new feature, anything published before that date is not guaranteed. If this is an issue, you can revert back to the original name for free at any time (but just once during the preview program). You will also be able to display your old ID along with your new one if you want friends to recognize you, however you can only enable or disable this feature when you make the actual name change - after that it's locked. The first change is free, and changes after that will cost €9.99/£7.99.
The rumor that PlayStation was making this long-awaited change surfaced earlier this month, when developers hinted they'd been told to prepare for the change. Once decided, you won't be able to switch between showing/not showing your old ID.
For the moment, it sounds like reverting back to your old PSN ID will be the only way to fix those issues, and considering that you can only revert back once during the beta test, here's hoping that Sony will figure out a better solution later on down the road. Sony will provide a list of supported ID change games (again, this is weird that it doesn't just work universally, and a glimpse into Sony's odd infrastructure).
Last week we heard from multiple developers that Sony was about to unveil its PSN name-changing service: something Microsoft rolled out many years ago that Sony fans have clamored for. Still, a list of compatible games will be available on the PlayStation site, so hopefully users will soon know which games to avoid.