You can definitely still use your MacBook Air, however the high noise speed can be very distracting and even annoying to have all the time, and will wear out the fan much sooner than otherwise.
I would take it into the Apple Store to see what they say. It's possible the SMC reset didn't really take. It's possible one of the internal sensors on the MacBook Air logic board have gone bad. It's also possible the CPU heatsink/fan has gotten loose (if it takes a minute or a few for the fan to kick into high speed).
You can download a number of free utilities to force the fan speed run slower/quiet, but the risk there is if the CPU activity is actually high and the fan does not keep things properly cool, this could greatly wear out the CPU life.
The first question is if when starting the MacBook Air from a total cold boot where it's been off for 15+ minutes at minimum, does the fan turn onto high speed almost immediately (even before macOS is loaded)? Or does it take so many minutes before it starts doing this?
Does the fan stay in high speed in recovery mode (booting with COMMAND-R)?
Does the fan stay in high speed while in Safe Boot Mode? (booting holding down the SHIFT key)?