I found something the other week and It's had a 100% success rate, However I'm not 100% Sure it's a Method or anything lol.
Right,
Step 1: Have 2 Different Browsers open with Twitter on.
Step 2: Make a Twitter you want the New Handle on.
Step 3: Change both of the passwords to the same thing and have it copied.
Step 4: Have the Twitter that you want to change it to make the word for e.g I have Graph so I would type Grap in.
Step 5: Go back to the one the Handle is on and Go as far to this:
http://prntscr.com/uvxkz Paste the password in.
Step 6: have both browsers up like this:
http://prntscr.com/uvyjs
Step 7: When You are ready Change the name on the one you want to swap and put something random like: elkdlf3333r
Step 8: Quickly goto your other browser And then Type : Graph with the "h" and It will be reserved, Then just paste the password in and Boom You have swapped your twitter, Someone else try this and please tell me if it works.
-Haunt.