Your approach to learning a song is as good as any. I use a similar approach.
Here's what I do a little different than what you described.
I usually start by just listening to the song over and over before trying to sing along. You may do the same thing as well, but I have found that if I start trying to sing along before I have heard it enough to begin to internalize the melody, I'm singing the wrong notes. And because I'm busy singing the wrong notes, I'm not hearing the correct notes as good as when I'm not singing the wrong notes on top of them. So I probably spend more time listening without singing along than I do singing along. I will even put the song on repeat play - and have it playing in the background while I do the laundry or the dishes or shave or shower etc. I will play it in the car while I drive. By the time I start singing along, I am only missing a few notes.
If I find that I am messing up on a particular section (the last line of the chorus for example), I will isolate that part using an "A - B" or repeat function on my mp3 player or on my computer and focus on just that one part until I have it down.
Another thing I usually do when learning a song I have never sung before, is download several versions of the song choosing those I like best. Sometimes the original artist will have alternate versions (although Elton John is not one to change things up so much). Many songs have been covered by other well known artists who always put their own take on it. Sometimes I can even find a really good amateur cover version on YouTube (search for "Song XYZ COVER"). So I will pick several that I like and download and listen to them all back to back. That allows me to choose the phrasing and embellishments that I like. I end up with my own version in the end. This is where singing along with one particular version would not work towards helping you craft your own unique version that still sounds authentic.
Many performers I know do their best to emulate the most popular version of the songs they cover and try their best to sound like the original artists. I know that no matter how hard I try to sound like Willie Nelson, or Elvis (Costello or Pressley) or Elton John, or Bob Dylan, or Johnny Cash etc. that I am never going to sound exactly like them. So I don't even try. I make it obvious that I have created my own unique interpretation of the song. This is easier to do when you accompany your singing with your own instrument as opposed to Karaoke.
Before I became proficient on guitar, I used to sing Karaoke. Like you, I found that many songs were originally recorded in a key that was too high. Most Karaoke players have the ability to change the key of the karaoke track. But I never thought about pitch shifting the song to listen to the lyrics as you suggest. That is a great idea.
Of course when performing the song with my guitar, I transpose to a different key. Another thing I do, is tune my guitar half step flat Then I can play the chords the original artists played (easier for certain signature riffs) but still sing in a lower key! Many artist such as the Steve Miller Band and Zac Brown have always played their music half step flat.
Now that I have learned to accompany my singing reasonably well on guitar, once I learn to sing the song by listening enough, I start practicing singing it with my guitar - and skip the karaoke altogether. One thing I do occasionally, is record my best take of a guitar vocal (me singing) and listen to that over and over. That lets me hear my interpretation from the audience perspective. When I do that, I usually end up changing something on the phrasing or change a melody note here or there. I use a multi track recorder so I can lay down the guitar part on a separate track. Then if I change the vocal, I can just overdub the new vocal on top of the guitar track that I saved.
That's the process that works for me. It sounds like your process works for you. If you like any of my ideas, feel free to incorporate them into your learning process. Otherwise, just keep doing what you are doing.
No matter how you get there, learning to cover a new song is very rewarding. And the learning process is part of the fun. I liken it to putting together a jigsaw puzzle. It gives you something to work towards and in the end you can sit back and admire the end result and say "look what I did" or in the case of learning to cover a new song "look what I can do".
Keep building your repertoire. And most of all - keep it fun!
