One issue is that you naturally hear via bone conduction, so what you hear is NOT only what is coming through the headphones. There's no way around this. You can increase the volume to try to drown out the conduction, but now you're risking hearing damage!
It takes time to become accustomed to hearing your voice from a recording. That's how others hear you, so maybe get someone else to listen to your singing, and then to your recorded voice to make the judgement on the tone.
Listening to your clip, I think your voice is ok. I hear a lot of room echo in the recording, which gives it a somewhat hollow sound. I don't know how you are doing the recording but if you are using two mics, then move the vocal mic closer. If it is cardioid, then you will gain some bottom end from proximity, and will increase the direct vs reflected sound. Using moving blankets or duvets can help dampen some of the reflections as well.
What is your current recording setup (mic, interface, etc). You might also look into using a real DAW, like Reaper or Cakewalk. It allows you to make adjustments to your sound more easily.