Well it's possible to do with a lower voice, too. Don't worry. I myself am a 'low' baritone.. Dickinson was one of the first singers I started emulating, at about age 15 or so. I speak from experience because this song was one of the first songs I covered, and I did the whole thing in falsetto pretty much. It sounded very strange, like a woman singing it lazily. Eventually I learned the technique is very, very different from that, but that doesn't matter.
There are people who use head voice/mix voice kind of technique and get a similar sound. From what I understand about the technique you are using, it is a process of strengthening. And that will happen over time. Just keep at it. Your cover is not bad at all.