Glenn - You eliminated most of the culprits when you said they also flicker when running solely off the battery. The most common cause is a bad (loose or corroded) 12v electrical connection. Starting with the easiest...if your coach has a battery disconnect switch, turn it off and on a few times to see if that helps. These switches are seldom used, so the contacts can get a little dirty.
Next, make sure the battery connections are tight and clean, particularly the battery ground cable, both at the battery and where it connects to the chassis.
Finally, check the connection where the positive battery cable connects to the 12v panel. If any of the 12v fuses are loose in their sockets, they can be tightened by removing the loose fuse and squeezing the socket prongs together, just make sure you first disconnect from shore power and disconnect the negative battery cable, so there's no 12v power at the panel. Good luck - Dave