Question: Q: MacBook Air Fan very loud ( have checked everything)

Do the following:

Install El Capitan or Later from Scratch

If possible, back up your files.

1. Restart the computer. Immediately, at or before the chime, hold down the Command and R keys until the Apple logo and progress bar appear. Wait until the Utility Menu appears.
2. Select Disk Utility from the Utility Menu and click on the Continue button.
3. When Disk Utility loads select the target drive (out-dented entry w/type and size info) from the Device list.
4. Click on the Erase button in Disk Utility's toolbar. A panel will drop down.
5. Set the partition scheme to GUID.
6. Set the Format type to APFS (SSDs only) or Mac OS Extended (Journaled.)
7. Click on the Apply button, then wait for the Done button to activate and click on it.
8. Quit Disk Utility and return to the Utility Menu.
9. Select Install OS X and click on the Continue button.

Is the computer still slow after the Erase and Install? Or did you mean the fan noise is still present? If the latter, then replacing the fan may work provided it's the fan that is noisy. I was also concerned as to whether the fan noise is the result of the fan running at high speed? If it is running at high speed, then the problem may be a bad temperature sensor since you already reset the SMC.

Try resetting the SMC once more. Reset the System Management Controller (SMC) on your Mac - Apple Support and Resetting your Mac's PRAM and NVRAM. Still no luck? See the following:

Using Activity Monitor

     Use Activity Monitor on your Mac

     Runaway applications can shorten battery runtime, affect performance, and increase heat and fan activity.

Using Activity Monitor Part 1- CPU and Memory

Using Activity Monitor Part 2- Energy, Disk, and Network

Also, run the Apple Hardware Test. It's possible that a sensor has failed and is causing the fan to run at top speed. You can also get one of the several fan control utilities and see if you can stop the fan from running at high speed always. But don't do this until you have eliminated a software cause using Activity Monitor.