Noisy Bath Fan

Best as I can describe, the bath fan noise sounds like a variation of continuous mild thunder. I pulled the fan/motor bracket out and plugged it in to a wall outlet and it sounded fine. (I'm not sure however if I positioned it the same as in the ceiling for this test.)

--The vent outside, the type which has 5 or so horizontal flaps, was about half open. Could it be partially blocked anyway?
--Overall the housing body of the unit seems not firmly secured to the studs--slightly wobbly.
--Lastly, as you can see in the photo, there are large gaps between the drywall and fan body.

I did check to see if the fan duct flap opens at the exit. I'd say it's acceptable, but could move more freely.
I found no blockage at the beginning of the duct work also.

What would be recommended at this point? Thanks!

 

I''d replace the guts, simple, cheap.

https://www.supplyhouse.com/Broan-C3...otor-and-Wheel

 

You can also get it at Depot and Amazon.
Broan-Nutone C350BN

 

Check your plastic squirrel cage. That thunder noise is often the hole for the shaft getting rounded out on the impeller. Most I've seen are supposed to be flat on one side.