Your tone isn't bad at all, what you need to work on is your intonation.
What do I mean by that? Firstly, let's define our terms more specifically; non-musicians often use "bad tone", "off key", "out of tune" and "flat" and similar terms quite interchangeably, but in musical language they refer to something specific, and different.
When musicians talk about "tone", "timbre", or "colour", we are talking about the sound quality of the note. For example, a trumpet, a violin, and guitar can play exactly the same note, and it will be the same pitch, but a different timbre.
Often the same instrument (in this case the voice) can produce different qualities of sounds, so for the same note on a guitar for example, you can pluck the string in different ways to produce a different "tone" or "colour".
When you sing, you have a pretty pleasing "tone" to your voice, it's not nasal, it's nice and open and "round" sounding. These are all subjective ambiguous terms of course, but that's perfectly natural when talking about tone, since it's something that's not so scientific to describe.
What you need to work on is your intonation. Whereas "tone" refers to subtle differences in timbre, intonation refers to small differences in pitch. Pitch refers not to the colour of the note, but how high, or low it is.
You are probably already familiar with the idea of different notes being higher or lower than others, but what you may be less familiar with is the idea that you can sing a note that is more or less the right pitch, but is just a little bit high or low. This is conventionally called being "out of tune", and more specifically being "flat" is being a little bit too low and "sharp" a little bit too high. If you're really out of tune then you might miss the note altogether and sing a nearby note, but even if you're only a little bit out of tune, so singing the right note but a little bit "flat" or "sharp", it can still sound bad.
The great news is that you obviously have ears good enough to detect that something is wrong, which means that you will be able to work on this. You might not yet be able to pinpoint the specific problem on your own, but that's just a question of practice and experience.
Confusingly, we sometimes we call people who can't hear pitch difference "tone deaf" (confusing, because it actually refers in musical terms not to tone (timbre) but to pitch. But hey, language isn't always 100% logical). You are clearly not tone deaf, your singing does follow the shape of the melody (just not quite accurately enough), and you can hear that something is off. These two things are an indication that your ear and brain know that something needs to be a little different; you're hearing the problem, you just need to train isolate it and fix it.
What's probably happening is that you're focusing so much on the "tone" and "quality" of your voice that you're not focusing on the intonation very much, which is the area you need to work on (because as I've said, your tone quality is absolutely fine, good even!).
Being able to recognise and reproduce small changes in pitch is a skill that must be practised like any other, and it's something that with singing is completely up to the singer (whereas with most (not all) instruments that have keys or frets or something the instrument does some of the work for you).
Well, now you know what you need to work on, it's intonation. Rather than write a long answer on something that I am sure has been covered on this site before, I will just point you in the right direction.
Instead of looking for "how to improve tone", what you need to look for is "how to improve intonation when singing" and "how to sing in tune". Also look for "ear training", which is the process of training the ear to recognise different pitches and reproduce them. There will be lots of questions on this site and elsewhere about these subjects and advice on how to practice them. I am sure once you start working on this specific area you will start to see some real improvements in your singing, and your nice vocal tone will be able to shine through on some beautifully in tune notes!!!