1. How do I sing a very different melody from an the tune on the instrument I’m playing at the same time – whether that’s piano or guitar. I often find myself relaying to the pitches of the instrument rather than singing an independent melody. I do practise a capella – but other tips would be greatly appreciated. By ‘singing an independent vocal melody’ my favourite example on youtube would be Orla Gartland – we really knows how to make her voice shine above her guitar, so the song covers a greater variety of pitch as a whole – adding more depth to the song. This is what I’d like to do!
2. I’d also like to develop some form of ‘vocal independence’. My best example of this would be Orla Gartland or Regina Spektor. I love their style and the way they can almost improvise pitches around a basic melody. Any tips of developing this skill would be awesome too!
When it comes to your first question, singing independently of your instrument until it becomes habit would probably be my best piece of advice. Then, put them together, focusing entirely on your singing, and none on your instrument (I’m assuming since yours singing is what’s being thrown off, your more comfortable on your instrument).
When it comes to your second question, the answer is simply do it. You must be willing to play around, without being attached to a certain outcome. If you hit a bad note, so what? I’d start by playing a single cord over and over and sing whatever vocal line comes to my head on top of it. Then, go ahead and play a series of cords, and create your own melody lines on top of it. Go wherever your voice leads you in the moment.
Side note, if you’re stuck in your head, judging every sound that comes out of you, you will not be able to do this well. So relax, develop a sense of non-resistance, and let go of your need to be right and just see what comes out.
Of course, mimicking to an extent can also be helpful, but don’t get too attached to their sound, because ultimately you want to create a sound of your own.
I hope this helps, and best of luck in your vocal endeavors!