You will get to switch from Democrat to an Independent party affiliation by registering again, but this time check "Democratic Party" from the list of political parties displayed on the voter registration form. After this you can now wait to receive your voter registration card in the mail.
I just did the same thing I went independent. In some areas the back of your voting card has the necessary form to change, you mail it in to your election office. Or you can ask for a new voter registration form.