With a 2-cycle, smoking does not necessarily indicate an oil consumption problem like it would on a 4-cycle because all the oil is in the fuel. What it usually means is it is burning too much fuel/oil mix. A mower that old might have mixture adjustment screws. Otherwise the float and fuel inlet valve may be stuck. Before freaking out, I would pick up the side of the deck with the air cleaner (engine turned off, of course) and drop it from about 6" a few times to see if it shakes the float loose. Start using a combination fuel system cleaner/fuel stabilizer like Seafoam, and add 2 oz/gal of Marvel Mystery Oil to your regular fuel mix.

What kind of oil are you using?
With Lawnboy oil being scarce I've been using Echo Power Blend for some time now in my 34 year old mower with good results. No smoking.
Also, do not use 2-cycle outboard motor oil. It will not withstand the higher temps of lawn equipment.

There is an "Altitude adjustment" needle valve that, if not set properly can cause smoking. It's on the left side of the carb when facing it. It's the only adjustment screw on that carb;. To set it properly, you close it, then open 1/2 turn. Start the mower and set it on low speed. Let it run to warm up a few minutes. Turn the screw a little either way to smooth out the engine. Then set the speed to high and, if the motor sounds like it's running lean or rich, adjust the valve until the engine runs smoothly. Also don't forget to check the air cleaner and as Sax said give it a drop to dislodge the float . Good Luck.