These printers have a lot of plastic pieces that can break with just the opening and closing of the front door. Some that lock the cartridges in place, others that actuate the cartridge shutter (drum cover). When they do break they can cause errors, jams, and even print quality issues.
For the 10.92.XX cartridge not engaged error there are a few different plastic pieces within the printer that might cause this error when they break. There are locking levers for both the left and right sides of each print cartridge. The left side locks are just mechanical as they hold the cartridge in the correct position. RC1-6633-000 for the color cartridge and RC1-7618-000 for the upper or black cartridge. The removal of the left side cover and then the HVPS is required to replace these. The right side locks are mechanical as well but also have a grounding spring involved. RC1-6640-000 for the right side lock lever and RC1-6641-000 for the grounding spring. The right side lock levers require the removal of the cartridge drive assembly. This procedure is much more involved than just a cover and a board so pay close attention when removing. If no error is generated when these fail there will most likely be some sort of print quality problem such as light print, uneven densities, and hazing on the page as the cartridges are not held in the correct position for the printing and cleaning processes.
There are other mechanical links in this puzzle as well. If you are having issues with the cartridge shutters not opening or closing completely when the front door is opened or closed you should then check the cam slide right and lock lever slide right (RC1-6645-000 and RC1-6643-000). These pieces help to control the movement of the cartridge engagement arm, cartridges shutters, and the OPC drive. The cam slide can be easily checked without taking off the covers by reaching over to the white rod arm on the right hand side of the ETB, and wiggling it. If it moves easily, the slide is most likely cracked or broken. To check the lock lever slide you will need to remove the right side cover for a visual inspection. If it is broken you will see it in the bottom of the printer. If it is just cracked at this point you should replace it.
13.20 error message where you cannot find any paper jammed in the printer or a 13.02 error message where a page is sticking out about one inch or so from the fuser. What might cause these errors beside the usual pickup roller or fuser replacement? You should check the pickup assembly. On the pickup assembly there are cams on each end with a small plastic protrusion that will lock them onto the pickup shaft. What might have happened is the left side locking claw may have become dislodged or bent and is no longer able to hold onto the pickup shaft. This can cause the pick roller assembly to move further to the right causing the registration shutter to stay in the open position. Once this happens paper jams or false jams will occur.