Your model isn't recognized at the Briggs site.
Sure it's not a 9T702???

IF it was only as old as you state, I'd let the dealer fix it under warranty.

Maybe the governor spring is off or the tab got bent trimming up against shrubbery etc. Distorted diaphragms will cause a rich condition.

Just double checked and had another set of eyeballs look too. It is in fact a 9T602. Unfortunately it's on a cheap-o mower purchased from a local Wal-Mart. I had one from there previously that ran great the entire time I had it so I wasn't worried about quality.
A friend of mine who has a lawncare business said that I should look into replacing the carburetor so I guess that's the route I'll be going.

Does it have mixture screws?? sometimes the vibration will cause the mixture screws to move alittle. IF you find mixture screw try turning it in 1/8 of turn or so. Just note the index of the screw head so you can turn it back if that don't help.

No mixture screws on late-model mowers. I'm guessing this one is a 'Pulsa-Jet' with the gas tank under the carb, and with the spring-loaded choke plate. I think the choke plate is sticking, enriching the mixture. If its like I say, take the air filter off and start it. After a minute or so, take a screwdriver and flip the choke plate wide open to see if it speeds up and runs right. The plate should move freely and has a little spring that holds it more closed. It works by staying almost closed at starting RPM so it enriches the mixture for starting. Once the engine is running at speed, the stronger air flow is supposed to hold it mostly open.
If this is the problem the choke area may need cleaning or other repair if the choke doesn't open after starting.