Looks like the sitting surface was fabricated with glued joints, where they left 4 corner notches for installing the legs (including the back) later. They then connected the legs to the sitting surface. These leg-to-sitting-surface connections are the ones that are probably failing.
There may have been glued dowels in addition to mechanical fasteners in these leg-to-sitting-surface connections. I would be weary of trying to disassemble things because the back legs are joined together above, so there's a good chance of 2 or more dowels to sever during disassembly. Instead you should probably add additional fasteners.
Hardwood like that tends to split pretty easily if you install screws without pilot holes. Chapter 12 of the National Design Specification calls for different pilot hole diameters depending on wood density and whether the loading is shear versus tension (they say "withdrawal" instead of "tension"). I believe that there's relatively little shear demand and that your wobble maps to creep or a tension failure.
In the case of creep, simply tightening the existing mechanical fasteners will solve the problem. I feel that this is unlikely the root cause for a professionally built piece of furniture like yours. It's an easy first step, though, so retightening is worth a shot.
In the case of a tension failure in the fastener(s) you'll need to install additional screws. Removing an existing screw for examination is a good method for sizing your new screws. Assuming there was one screw, do two screws: one above the failed screw and one below the failed screw (if there's not enough space, then you can alternatively install a larger diameter screw in place of the failed one, but this presumes a pull-out tension failure instead of a splitting tension failure).
For a hardwood with oven dry specific gravity between 0.5 and 0.6 (like maple), NDS 12.1.5.2 prescribes a pilot hole with 70% of the diameter of your screw's minor diameter (the solid part's diameter, where you ignore the diameter across the tips of the threads). If the hardwood's oven dry specific gravity is greater than 0.6 (like oak), then NDS 12.1.5.2 prescribes a pilot hole with 90% the diameter of the screw's minor diameter. These pilot hole diameters will maximize the screw tension strengths, so be sure to get it right. When installing you want to just draw everything tight and then take an extra 1/4 turn (I couldn't find an authoritative source, so this "1/4 turn" is my best guess--authoritatively it's 1/2 turn for Douglas Fir plywood for what that's worth). Overtorquing will greatly reduce tension strength, so I would use an old fashioned screw driver.
Feel free to lubricate the screw with soap or wax--anybody wringing their hands about corrosion is full of it. The last thing you want is to twist off the screws before installation is complete. And shop for screws calibrated for hardwood. They tend to have finer threads than the softwood screws.
