I sat in line for about an hour and a half to get a new driver's license with my new address on it. They gave me a paper copy of a new one and told me that I would get the real one in the mail in awhile. They did not take my old one, which I am still carrying.
I called the state patrol office and asked them what I needed to do for my chp. They told me to download the same form that is used to apply for a new one, fill it out, and check the box that says change personal information (i.e., address in my case), and mail that form to the Lincoln office with a copy of my new driver's license. They did not tell me to send in my permit (presumably they already know I have one).
So unless I was given bad information, I can keep my old permit and driver's license with matching addresses until I get both of the new ones. This is a more efficient process than I had expected (except for the long wait at the motor vehicles office).