I had this problem before...
2008 Panel LT2
93k+ miles
When braking between 55 and 50 mph I get a slight shimmy for a few seconds, shimmy is more severe as more pressure is applied. Last time my pads were worn so I replaced them along with the rotors. Still have plenty pad left but the shimmy is back and still only at around 50 to 55 mph.

Thanks....

 

Check your lower control arm bushings! If they are toast you'll get the shakes!

 

Did you clean and adjust your rear brakes? The "Self Adjusters" on HHR's are infamous for not self adjusting worth a darn.

Most of us "Old Heads" here, clean and adjust our rear drums at every oil change or every 5,000 miles. If the rears aren't doing their 30-35% of the braking, your front brakes are constantly being overworked.

https://www.chevyhhr.net/forums/brak...ocedure-36754/