I have exactly the the model (FTW445GES1) showing the same problem.
Also when I press the door, the wash machine starts to work.
Today a repair man came and quoted $270 to replace the door lock
assembly.
I could not decide whether it was a good idea to pay for the amount.
I am now contemplating whether to have it repaired or buy a new one.

I would appreciate what happened to your wash machine afterwards. Has
it been repaired and how much was the cost?

SK