Black tanks can be quite shallow. Ours in only about 6" high. If your broomstick won't go in far it's probably hitting the tank bottom and won't tell you much.
Liquid Drano can be safely used. In fact, our owners manual says to use it. Won't hurt metal or plastic but double check the label if you buy another brand. You'll need to drive around to let the Drano + contents slosh around. May need to repeat multiple times to get the tank thoroughly cleaned. I use it once or twice a year to get the tank back to factory fresh, factory clean condition.
I see that your unit is an '06 Raptor. Is this a sudden problem of getting worse over time? Have you owned this long? With black tanks, you need to add a few gallons of water before each use and you shouldn't skimp on water with each flush. If you don't, you can get the infamous pyramid of poo and at the least, the tank below the toilet can get pretty gummed up and "stuff" won't move along very well.
In some cases, the factory may have left behind a disc of the ABS inside the tank but on an '06 would be unlikely. If you hold the toilet valve open, water *should* be able to freely drain out through the outlet. If the valve is remote cable operated, need to see if the blade valve is opening properly. Not easy if you have an enclosed underbelly.
Before dumping, you always want to fill the tank to full or near full. Forget about using the sensors to see how full the tank is. Shut the water off and peer down the toilet with a flashlight. I added a shutoff valve behind our toilet to make this easier.
Tanks come in all sorts of dimensions, configurations and outlet locations. With some, pushing a snake or hose back up the outlet won't do much but is worth trying. I once used about a 6' length of stiff rubber garden hose back into the tank to clear it. You could also try pushing a hose around the tank from down the toilet. With some tank configurations, a backflush elbow on the outlet won't do much - doesn't work on our current TT but worked great on a previous TT. A clear elbow is great for seeing what's coming out and how clear the water is getting.
If you're at a CG, see if they have a honey wagon service. They may be able to suck the tank clean.