Q: Engine timing

Car worked fine no lights on dashboard night before. Next day it’s cold out I let the car sit and warm up all of a sudden car shakes and power cuts off I turn it back on it does the same thing and engine light starts flashing. We put fuel cleaner in the tank and drove and lot of white smoke came out the tail pipe and it began to smell like rotten eggs we tried to drive it around and the engine light became steady but it ran a little better. The next day the car didn’t start at all. The tow guy said it sounds like my water pump failed because he smelled gas when trying to start it he said the car wasn’t getting any gas and pepboys mechanics are claiming I need a whole new motor because the timing is off. No one checked for the fuel problem. My question is does this sound accurate? Could the timing problem be fixed without ordering a whole new engine? The engine is still getting a spark they said it’s just the timing.

My car has 87000 miles.
My car has an automatic transmission.