Welcome! This forum has a treasure trove of great info – Scouters helping Scouters! Just a heads up, though - all content, information, and opinions shared on this forum are those of the author, not the BSA.
There have been previous requests to add adult awards to Scoutbook. The BSA has previously stated these are prioritized below items that directly affect Scouts and/or unit operation.
Wood Badge is recorded (or supposed to be) in your training record at my.scouting.org. If it is not, contact your council or district training committee.
Scoutbook at one point did support the adult training completed. But, sometime ago, the Scouter trained report was flipped to the Training Roster it my.scouting.com site.
I feel your pain, I had uploaded all my train including copies of the c completed cards and certificates. Now they are lost in cyberspace.
Admittedly the training has always been tracked as a matter of record in my.scouting.org and scoutbook allowed it it be entered there but did not track to the official record. A key 3 or council training chair can enter them if not shown.