The only saving that Super Mario Bros. 2 does is the star count on the title screen, which goes up to 24. When you get 8 of them, you can access world A by holding the A button on the title screen and pressing start.
Most of the big emulators (Nestopia-UE, Mesen, PuNES) all do FDS stuff just fine. I suspect you're running a hacked cartridge port of the game rather than the original famicom disk system disk image. What's the MD5?
I haven't played my Switch n a month or so, just been playing Xbox, and just picked to play lil and Damn that A and B, X and Y being reversed always gets me.
@BigOnYa:
I haven't played my Switch n a month or so, just been playing Xbox, and just picked to play lil...