I applied for my name change within a month after I went full-time. I had some paperwork hassles with it, so I didn't get the certificate until three months. In the intervening time, I didn't have any problems.
I did have to produce my bank card with my old name on it when I went into the bank one time. I was expecting to get questioned on the name, but it was actually my age that was suspicious: she didn't believe I looked 62. Fortunately, she took my word for it that I was, because my next step would have been to produce my driver's license, with an extremely unconvincing picture of my old self. For the name, she just offered to enter a preferred name into the computer.
Since you are already full-time, there's not a lot to do except apply for the name change, and avoid situations that require you to produce identification. Don't speed or run red lights!