I agree that $350 is way excessive. I winterized with pink stuff for 15+ years (now I just take the fiver to Arizona) in Northern Illinois winters with never a problem.
First you bypass the hot water heater (maybe they had to put in a bypass to justify that cost?);
drain the water heater;
drain the fresh water tank;
if not there already put a T fitting in the line to the input of the water pump with a short hose attached and a couple of valves to switch between the tank feed and the short hose;
drop the hose into a gallon jug of antifreeze;
turn on the pump;
go to the faucet farthest from the pump and run cold water it until it runs pink (nice to have a helper out by the pump to switch to a new jug occasionally);
then run the hot side until it runs pink;
repeat for all of the faucets, shower and toilet;
disconnect the line to the shower head so it drains;
don't forget to open the check valve on the city water connection and let it run pink as well (try not to get a face-full of water - don't ask);
running the taps will fill the traps at the same time and put a little bit of anti freeze into the gray and black tanks, which should have been emptied as well.
Don't forget things like washer connections, sprayers in kitchen sink or toilet, lines to built in water filters (which should be drained & bypassed), ice makers, and outside showers.
Whole job should not take more than 45 minutes and you know it was done right.