Escalation approval tool, copy to clipboard via Python
The main part of your program is only

def copy_string_to_clipboard(self):  # Copy to the windows clipboard
    cb = Tk()
    cb.withdraw()
    cb.clipboard_clear()
    cb.clipboard_append(self.full_string)


the rest being used only as a formatting tool. You should thus focus into making such formatting as clear as possible.

But first of, you may want to read about docstrings as your comment should be one:

def copy_string_to_clipboard(self):
    """Copy to the windows clipboard"""

    cb = Tk()
    cb.withdraw()
    cb.clipboard_clear()
    cb.clipboard_append(self.full_string)


So now, onto cleaning a bit everything around this function.

First off, you don't need a class with 3 other functions just to concatenate two strings. Based on your opening variable, what you need to do is simply:

user = read_username()
assignation = raw_input('Escalate to where? ')
copy_to_clipboard(
        'Ticket has been reviewed and approved by {}. '
        'Ticket assigned to {}.'.format(user, assignation))


where copy_to_clipboard has been slightly modified to accept the whole string as argument:

def copy_to_clipboard(text):
    """Copy the content of a string to the windows clipboard"""

    cb = Tk()
    cb.withdraw()
    cb.clipboard_clear()
    cb.clipboard_append(text)


The last thing to do is to simplify a bit the reading of the user config file. It's a bit of a shame to be required to run the program twice in order to create a file if it doesn't exist. Instead, you could try to read the file and, if it fails, ask the user for its content:

def read_username(filename='esc_name.txt'):
    """Get the user's name from config file or command line"""

    try:
        with open(filename) as f:
            return f.readline()
    except OSError:
        pass

    # If we are here, we couldn't read the configuration
    # file so ask the user an try to save its choice
    username = raw_input('Enter what you want appear on escalations (be specific): ')

    try:
        with open(filename, 'w') as f:
            f.write(username)
    except OSError:
        pass

    return username


But this function assumes we are in the right directory, which may not exist yet. Let's tackle that as well:

from getpass import getuser


def setup(program_dir='esc'):
    """Create our own directory structure and move to the right place"""

    appdata = r'C:\Users\{}\AppData\Roaming'.format(getuser())
    os.chdir(appdata)
    if not os.path.exists(program_dir):
        os.mkdir(program_dir)
    os.chdir(program_dir)


as you can see, I used getuser from getpass directly without having to create a wrapper that just remove 1 level of nesting.

Also, if I recall correctly, os.environ['APPDATA'] should contain the value we are building into appdata. So you could simplify it to:

def setup(program_dir='esc'):
    """Create our own directory structure and move to the right place"""

    os.chdir(os.environ['APPDATA'])
    if not os.path.exists(program_dir):
        os.mkdir(program_dir)
    os.chdir(program_dir)




Putting it all together, you can end up with:

import os
from Tkinter import Tk


def copy_to_clipboard(text):
    """Copy the content of a string to the windows clipboard"""

    cb = Tk()
    cb.withdraw()
    cb.clipboard_clear()
    cb.clipboard_append(text)


def setup(program_dir='esc'):
    """Create our own directory structure and move to the right place"""

    os.chdir(os.environ['APPDATA'])
    if not os.path.exists(program_dir):
        os.mkdir(program_dir)
    os.chdir(program_dir)


def read_username(filename='esc_name.txt'):
    """Get the user's name from config file or command line"""

    try:
        with open(filename) as f:
            return f.readline()
    except OSError:
        pass

    # If we are here, we couldn't read the configuration
    # file so ask the user an try to save its choice
    username = raw_input('Enter what you want appear on escalations (be specific): ')

    try:
        with open(filename, 'w') as f:
            f.write(username)
    except OSError:
        pass

    return username


if __name__ == '__main__':
    setup()
    user = read_username()
    assignation = raw_input('Escalate to where? ')
    copy_to_clipboard(
            'Ticket has been reviewed and approved by {}. '
            'Ticket assigned to {}.'.format(user, assignation))
    print 'Press CTRL-V to paste the approval.'