I have tried via some blogs, have joined mailing lists of W3C and WHATWG.
How should I proceed for writing spec for a new HTML tag and subsequently propose it to W3C?
My current employer is a member of W3C.
In literal terms, just write it and post it, and make it known via the mailing lists.
Writing the spec is the easy bit. Getting implementers to buy into it is the hard part. You have to convince them that putting the work into implementing and maintaining it will make for a better web, more so than the other things that they could be doing. That's a huge hurdle to overcome.
So it's actually more useful not to start by writing a spec. Instead post to the lists your description of the problem that you're aiming to solve and try to convince the others there that the problem is one worth solving. If you can do that, they will then be able to help you identify the best way to solve the problem, which may or may not involve the creation of a new element, and also help flesh out the use cases for the solution whatever it happens to be.