Stuff from the default UI and probably settings of addons that save per character are transferred, yes, but many addons use global profiles to store settings - mainly so you can manage them on any character, copy them to new characters etc.
For these, I recommend e.g. Notepad++, a text editor which allows you to search across whole directories instead of just one file. There are certainly other programs that can do that, too.
After the transfer, preferably before logging in, search for your character's name (the old one, if you changed it) in the
\World of Warcraft\_retail_\WTF\Account\ACCOUNTNAME\SavedVariables folder. Use *.lua in the file filter, ignore .bak files.
I advise to back up the whole folder (read: copy elsewhere) before doing this, so mistakes are less severe.
For the most part, the name will come up as a profile index, something like:
Change the realm, as well as the charname if you changed that.
Note that this might occur more than once per file.
Occasionally, the addon uses the realm as a higher-level wrapper, so your character would be alongside your other characters from the old realm. This should usually just be addons that collect data about the character, e.g. inventory or boss kill counts, automatically re-created for the "new" character, so make a note of this, and maybe purge the old data later, using the addon's ingame methods.
If the data is important, you can move this, too, but be mindful of the syntax, most importantly the curly brackets, and I suggest to log into and out of the transfered char first, to let the addon show you exactly where and how it would put the data.
Notepad++ knows the Lua syntax and lets you collapse a bracket that covers multiple lines e.g. data for a character, so you can simply cut that one line and get the whole data for the character without having to worry about the brackets: