All I'm trying to do is understand the possible causes and options to fix the problem. Whenever I do repairs, I always start with the simplest possibility first, so of course I'll change the bulb before doing anything else.
If you are sure the signal lamps on the offending side are good, then tracing for good ground is the next thing you should be doing. And it's not at all difficult. If the car in question is the 2011 Camry, it's very possible you could have an issue with the harness ground(s) as a result of age, vibration, or even corrosion.
While I'm not familiar with the Camry's specific harness colors, this info should be easily obtainable online.
The turn signal sockets in question should have three wires: One 12V feed for the parking lamp, one intermittent 12V feed for the signals/hazards, and a ground. The key kicker is to discover which one is the ground (wiring schematic or multi-meter). On my Dodge products, the grounds are black with green or brown stripes. Can't say for Toyota.
The socket wiring undoubtedly disappears into a harness/bundle, which can make specific diagnosis of an individual ground wire quite difficult. However, the bundle itself should have a ground wire emanating from it someplace along the body shell. On my old Caravan, for instance, the front harness ground point is a small stud with a nut on the inner part of the radiator upright. If you follow the socket wires into the harness, and then follow the harness, you'll find the ground point (stud). Make sure the ground wire is intact and it's all clean & shiny metal (not corroded or full of gunk).
In the case of the last one of these I fixed, the ground point was very clean and was therefore of no help. And since the ground wires (right and left side) were buried in a bundle, I really didn't want to start dissecting.
One way to check this theory is to run a separate ground wire to the ground wire of the socket(s) in question. You'll have to peel back a tiny bit of wire insulation and connect a jumper from that point to a clean ground point (I scraped paint from a hidden part of the body behind the headlamp). Once sure you have a good ground, test the bulb and its functions. If it works, you probably have a broken or corroded ground wire (inside the harness).
This sounds wordy and perhaps challenging, but it's not. 1) determine ground wire; 2) check the harness ground point for breakage or corrosion; 3) run a separate ground to the socket in question.
If it's in fact a bad ground wire, just make the jumper wire permanent. Here are some pix from a job I just did to fix this exact issue. I'm not saying this is your problem...but it's a possibility you should investigate nonetheless.