If it was the PTT motor all you'd get is clicking on the solenoids when you trim up/down, normally when the motor fails it fails, end of...and even then it's the brushes which need changing. It's normal to have corrosion on the housing. Saltwater.
Give it a light hammer test and that will show if it's beyond service. You can replace the casing if needed but not really worth it better to get new PTT motor Instead.
Like others have suggested you can eliminate the PTT system but disconnecting it from use.
My bet is bad earth somewhere. Finding it is a whole different story. Be careful what wires you wiggle when running though :-)
Not sure if your motor has the diagnostic plug in. May help pick up faults to get I plugged in and checked.
Peter ~ firstname.lastname@example.org
~ 07930 421007