If the starter was staying engaged it's an issue with the solenoid.
My friend still has his shop down on 30. I think you may have been there once. He may have a good used 1 that m8ght get you by.
Pn# 7c3z 11002-aa , check to see if they are quoting you a remain suffix If I remember it would aarm, arm or Abrm and not -aa, should bring the price down.
471.00 + 2 hours labor. Is what I'm seeing. Sorry sir. Better to be forewarned