I am going to guess that you might be listening to the vocal track and trying to follow it exactly as it occurs (in real time) instead of memorizing the notes ahead of time. If that is the case, you may be adjusting on the fly as you sing along to the vocal track, causing you to be a little off balance and not sure where to place your voice because you are not quite sure what the next note will be (until you hear it).
If I am trying to learn to sing a cover and sing the same notes the original artists sings, I go through a process to learn the song so I can sing it from memory. Then I can sing along with the recording and I know what note I am going to sing next (before I sing it) so I am not really listening to the vocals - but cuing off the instrumental music instead.
I do this step by step, line by line. What I do is listen to the first line of the first verse multiple times and then sing it back to myself and then along with the recording. I usually miss a note or two the first few times so I keep trying until I have the first line of the first verse down pat - note for note.
Then I repeat with the second line by itself until I have it memorized note for note and then put those two together. The process builds from there through the entire song. Once I have the song memorized, I don't have to listen to the vocals to sing along with the vocals because I know where they are going before they get there (no adjusting pitch midway through a line).
If I am doing this at home, I use the computer to play the recording with a visual display on the screen - making it easy to skip back to the point I want to repeat. If in my car, I use a digital handheld recorder plugged into my auxiliary jack on car radio and I can set an A to B repeat on the recorder.
By the way, some songs don't have complex melodies and are easier to memorize without going through the process described above.
Of course, when I am playing the music myself on guitar, I don't have to memorize the song exactly and I often sing it using my own "interpretation" (a fancy way to say I am too lazy to try to learn it exactly the way the original artist sings it).
You might try my method and see if it works for you. The main thing is that you must get to a point where you know what note is going to be sung next - without waiting to hear it first. Otherwise you will be like a batter in baseball trying to hit a knuckle ball (swing and a miss more often than not).
If you memorize the melody ahead of time - it won't matter how loud the vocal is in the mix and you should be able to hit the notes even if the vocal track is suddenly muted. The instrumental music should be all you really "need" to hear to sing along with the vocals on the record.
Good luck and keep it fun.