The factory disconnect should turn off the inverter when set to off. My CONverter was wired backwards in my 2019 Eagle TT and the disconnect did not shut off power to the coach and even 2-3 nights without shore power would result in a dead battery. There should be two heavy gauge red 12V wires in your converter. When disconnected, one should have power and the other should not. When wired backwards, 12V power was available to the coach AND the inverter.
Assuming similar wiring, your solar should be wired to a bus bar on the "in" side of your disconnect switch, so it should charge the battery regardless of the switch position. That being said, do you have a solar controller? A controller will maintain your battery. If the panel is directly hooked to the battery, it could potentially overcharge and damage the battery. Really depends on how big your panel is, but it isn't intelligent to "trickle charge" your battery.
Both situations are best tested with a multimeter. With the disconnect off, you should only have 12V on one leg of the switch. On my rig, the only things that are connected directly to the battery side are the one converter line and the break away switch. Even the tongue jack is off. I'm again assuming you don't have a solar controller based on your post, in this situation you can easily test the solar line by checking for 12V between the + and - at the panel (with the panel disconnected). Since it should be wired directly to the battery, you should have 12V at the solar in jack (roof or side). If you do have a solar controller, then your controller ought to tell you what's wrong from the solar side.