Blue (or white?) smoke indicates oil consumption. Black smoke indicates excessive fuel consumption (too rich). So if the smoke is not black, then it shouldn't have anything to do with the carburetor.
I don't know what kind of condition the mower is in but if you've had it that long, it might just be worth replacing the entire mower even if you were mechanically inclined.
I've offered several disabled push mowers to a guy I work with who works on mowers in his spare time and he says fixing up free push mowers are not even worth his time fixing and reselling. The only time he repairs push mowers is when people who bring them to him for repairs.
Opinions may vary but Briggs and Stratton are notorious for oil consumption after they get a couple hundred hours on them. I bet my 21 hp riding mower has less than 500 hours on it and it uses about a quart between oil changes (every 100 hours). I even replaced the head gasket on it (which was blown). In this case it was fouling the spark plug so I had no choice but the replace it. Every B&S engine I've owned burns oil. Kohler makes the best engines, in my opinion.
I used to have a Snapper tractor type with a B&S engine and for some odd reason it never smoked, leaked oil or fouled the spark plug but consumed a quart of oil every time I mowed (3 to 4 hours). Instead of putting brand new oil in it, whenever I changed the oil in one of my cars, I would just drain it into a clean pan, pour it into a jug and top the engine off before mowing. I did that for years until I sold it.
If your engine is not fouling the spark plug, I would just top it off with used oil and not worry about it.
Used engine oil will smell like gasoline even if the engine is OK. That's just normal. If the oil level is rising, that would indicate that the float valve is leaking and filling the crankcase with gas. But if you are having to add oil to the engine, then that's not the case.
You could also find a used mower for free (a the dump for example) with a good engine but cracked deck. Then transplant the good engine over to your good deck.
As much as replacement engines cost, and as much as replacement parts are, I'm not so sure if I would even rebuild a lawnmower engine. Back in the 1990s, I was going to rebuild a 12hp B&S engine that had seized. By the time I added up all the parts, I could have bought a new one! I usually easier and cheaper just to find a lawnmower with something else wrong with it.
My father gave me this old Yardman push mower. It has a good Honda engine on it and uses no oil. The plastic piece that raises the front wheels had broken and the part was $80. So I just leveled the deck and welded the bar to the deck. Otherwise If I didn't know how to weld (well somewhat), would have junked the mower.