There are some great resources over at Travel Oregon on ghost towns. You can also find helpful details at Visit Eastern Oregon here. According to these sources and others, it does appear that Auburn is fairly rundown and collapsed, with little signs other than rubble and gravestones from the old town cemetery that can be easily missed.
Other explorers suggest you continue past French Gulch Road where you will reach a split in the road, and if you stay right for a few miles you will approach the old town of Auburn near Freezeout Gulch. These roads are considered rough and often require four-wheel drive depending on the conditions. I have read that that the area is both private or public based on how far visitors venture from the road, so it would be useful to stop in at the Baker City Chamber of Commerce and talk to the local experts if you can. Hope your adventure to find these historic spots goes well!