I am sorry but the service manual for this model is not available. I hope that the following can help a bit.
38HN-3TXW are listed at appliancespartspro (here the link)
. As you know parts for the control panel are not available.
Parts for similar models are available. I would contact them to find out if parts from models like 38HN-6TVW-SC
Here diagram of 38HN 6TV control panel,
it is similar to 38HN 3TX, shown at partselect.com
(that shows parts as discontinued).
Here a list of models from 38HN series
A way to troubleshoot without service manual is running continuity tests between contacts at switch, relay board, control module and elements to understand how wiring is connected and to test wiring. In a second stage the job consists in reading voltage at relay board and timer contacts. Because of the safety issues involved, testing with power on must be done by a trained person.
Where possible do continuity tests. About the corroded contacts at relay board check if there is continuity between the solder joints at the back of the board and the relay connectors (continuity means contact closed with zero Ohms or low impedance). The test is done with power not connected.