The flashing light has several causes, all to do with fuel supply or the fuel system's evap function.
Check the fuel pressure before you go any farther. ANYTIME the engine is running, and for a few seconds after you go to "ON" with the key, even if the engine does not start, you should see 55-62psi.
Even 40psi is a problem, and under 30psi, the car will do little more than idle speed.

Keep in mind that finding the fault is the hard part, once you find the fault, fixing it is usually rather easy. Most garages do no better at finding the fault than you will, because diagnosis and troubleshooting are not effectively taught these days.