riddle - New yet ubiquitous ... what am I getting at?

With eight, I'm quite heady.
With three, my claws ready.
With thirteen, I'm downtrodden.
With eighteen, said to be sodden.

Alone, I'm quite new here,
Yet ubiquitous everywhere.

What am I?

Answer

The numbers are referring to

Letters of the alphabet!

You are

The word 'at'

With eight, I'm quite heady,

Hats are worn on the head [citation needed].

With three, my claws are ready,

Cats are known to have sharp claws.

With thirteen, I'm downtrodden,

Mats, like doormats and floor mats, are walked on.

With eighteen, said to be sodden.

This one could be a reference to the phrase "wet as a drowned rat".

Alone, I'm quite new here,
yet ubiquitous everywhere.

Thanks to Ryan, I now know that it refers to the @ symbol! It's new to the common keyboard (relatively), yet is in common usage on pretty much all keyboards, websites, etc. these days.