Macbook Air Lagging after upgrade to 10.12.6

Downgrading macOS is never justified for the symptoms you describe.

Please refer to the screenshot in See how apps affect Mac performance, battery runtime, temperature, and fan activity - Apple Support. Follow Steps 1 and 2. Determine if you can correlate a process or processes with your Mac's degraded performance. Identify those processes, or post a screenshot of your own. To learn how to post a screenshot, edit it if necessary, and post it to this site read the Appendix in Writing an effective Apple Support Communities question.

If you are unable to correlate any particular process to the degraded performance and increased fan speeds, then follow these instructions excerpted from About fans and fan noise in your Mac - Apple Support:

"If the fans in your Mac run fast even when the computer isn't experiencing heavy usage and is properly ventilated, you might need to reset the System Management Controller (SMC) on your Mac."

Just to be clear, is there a PC involved, as in a Windows PC? Are you connecting your Mac to a PC?

Post the screenshot anyway. It may reveal a suspicious process. Please edit or obscure any personal information before posting the screenshot, if there is any.

Consider running Apple Hardware Test / Apple Diagnostics: How to use Apple Hardware Test on your Mac - Apple Support. A report of a specific failure would be useful. "No trouble found" is not particularly useful.

thainan10 wrote:

According to the test, may have problems in the SMC, power management and camera, but it just appeared after the update and these three things are seemly working right.

That's not a result I would have expected. It can certainly explain the problems you've been experiencing.

Some "problems" reported by AHT may be spurious. Please run it again with the Mac's power adapter connected, and describe the problems it identifies.

Please Contact Support to have that Mac evaluated for a possible hardware fault. There are a few exceptions for a "No trouble found" conclusion, but the results you describe are not normal.

The fix may be simple and inexpensive and possibly even free, but the only way to be sure is to have Apple evaluate it.

I seem to have the same problem Macbook Air runs unbearably slow after update to OS 10.12.6

Can we exclude software interactions. Were you using apps such as Dr. Cleaner or ExpressVPN?

Probably Spotlight is reindexing your Disk. You can use Activity Monitor to verify Disk Activity and cpu usage. Leave your mac turned on until Spotlight has finished.