Summary – each Twitter account has one Twitter ID. Changing the screen name does not change the ID. Tweets are sent to the ID, threaded conversations help to find new name.
If you change your @-name on Twitter people will still be able to find you, from your original @-name. This is because your Twitter ID stays the same regardless of what your screen name says so any message sent from or to your account will immediately inherit your new name. It’s easy for someone to look at the tweets sent to you and find your new screen name, so a name change does not work as a stealth move on Twitter.
Deleting all of your own messages won’t help (and you can’t delete the ones sent to you anyway).
To find your current screen name someone could search for a tweet sent to your old name, click on it to bring up the thread and it will now display your current name in the bit where it says who it’s sent in reply to. This only works for a tweet sent to you (as first recipient) as opposed to one where your name is mentioned later in the tweet.
An example of a search would be from:personA @oldname where personA is someone who’s definitely sent a tweet to an account under its previous name and @oldname is the old screen name. (NB using to:oldname won’t work as Twitter now automatically updates the “to” bit. I found a tweet I’d sent in 2009 to “oldname” and it now says I sent it to “newname” but the old name appears as text in the tweet).
If you want to be unfindable you’d need a different account I’m afraid, I don’t think there’s a way around this.
At some point I expect the change Twitter’s made to the way in which names now don’t quite show up in replies (no longer included as part of the text or character count) may mean that people will better be able to hide themselves in future against such searching, but possibly not.
This is adapted from an answer I wrote on Quora. It updates an old 2014 post of the same name that I wrote on my other blog.