As mentioned above clean & lube the shift pedal pivot point.
The gear shift springs are on the right side behind/below the clutch assembly.
Shifter rod goes across from right to left.
The other part is the gear shift drum more to the rear. Parts and pic from svc manual.
Buy OEM Parts for Honda Motorcycle 2005 Gearshift Drum Diagram
www.partzilla.com
It was not until 2006 that the 14 pin yellow connector with Ground wires was on a schematic. Complete schematic Right half and left half each can be printed and joined.
www.vtxoa.com
Where are you?? Finding a good tech is difficult..
Some dealer just blew up a VTX drive train.