Hello MRedstone
I will give you my advice below here with some description included;
There are plenty of neat recording/streaming softwares which you can use and is compatible with fullscreen mode and more. The program I will recommend for you is OBS (Open Broadcaster Software). OBS is maybe the most used free recording and streaming software with tons of great functions suiting almost every need. You can currently download two types of versions of OBS and that is OBS Classic and OBS Studio/Multiplatform.
First of all, here is the website where you can easily download OBS; https://obsproject.com/
I would recommend that you use OBS Studio since OBS Classic is no longer in development and Studio is the version now they work on.
With OBS Studio you can record ETS2 easily with full screen and even add some fancy effects or text in the screen if needed. So after downloading OBS Studio, head over to YouTube or whatever video sharing platform you prefer and search for tutorials. You can find tons of easy and helpful tutorials for beginners who have never used it before. And of course, there are lots of guides and discussions about OBS Classic and Studio in forums, websites, etc.
So that's my advice for something free, easy and with quality. And I also use it myself :3
If you have any questions, I will happily try to answer them
Thanks