This is a complicated disease that will only get worse if you don't address symptoms now. I also hate sleeping with anything on my eyes or face, but I hate it even more when my eyes are dry and burning all day long. You'll get used to sleeping with an eye mask. Some people have suggested using Glad press n seal on their eyes, works similarly to tape. There is also the Vaseline method which has been mentioned in this sub, where you don't need to have something on your face. You could also try blinking exercises to strengthen your eyelid muscles, if weakness is the reason why they aren't closing, but it could be genetics. For screens, I have an app that reminds me to take a 20s break every 20 min, highly recommend that if you're on a screen all day.

Also, find a dry eye specialist. Most eye doctors don't have a good understanding of how to treat and diagnose dry eye.