Chevy cobalt 09

So I’ve had my Chevy cobalt for about 4 years,
never did the engine light turn on. Today it did, I
went to autozone they told me I had to replace
Engine Control Module, but to make sure it was that
I should go get a second opinion. Wen to a
mechanic and he said I had to replace the timing
belt. My engine shakes when I am at a stop but
runs good. It’s being like that for a while now, also
when I’m running 65 mph or more and I press on
the brakes, car staring wheels shakes. But only
when I press on the brakes and I’m going fast. I’m
not sure what is happening here. Can anybody help
me out understand better? Thanks in advance!!