I have a 2001 honda accord value package., which had two broken motor mounts that I saw were broken with my own eyes. So yes it is possible you do have broken motor mounts. The front one is easy to see. The rear one has to be looked at from underneath the car. Good luck. OBTW my mechanic said the rear one was a bear to put in, so expect a pretty penny to be charged depending on who does it.
I had 212,000 miles when replaced.