How can we drain just the grey water tank on our 2014 Dutchman Denali 287RE without dumping the black water?

Sometimes these valves are identified by the color. Grey for grey and black for black. Another way to tell is that most of the time the black fittings and pipes are larger than the grey. Look at your plumbing underneath the unit at the dump valves. The valve that accepts the larger pipe will most likely be the black.
This is a good illustration of the two valves. You can see that they are color marked, and that the black valve piping is much bigger than the grey beside it.

the grey water is your shower and sink drain tank. the black water is your... you know. the two combine to the dump that's held together with a bypass for grey water to flow into the black tank and vice versa so you can flush the black tank further. all of this comes out the end into your dump out for black and grey. If your talking about white water (potable) holding tank, filling this drains through the use of it, run the shower outside or hold flush down until it drains. otherwise. the hotwater tank will drain with the anode when you regularly change that once a year as directed by people who camp once a year - change that anode -

There should be a compartment on the outside, typically where most utilities are connected, that will have the drain connectors for black and grey water. Black is for your toilet, grey is all other water water. Look for the labels to ensure you have the correct one. Depending on the manufacturer, it will be either a black, plastic or silver, metal T-handle. Once located, you will pull the T-handle out, approximately 4-6 inches (10-15 cm).

You'll probably have to crawl under and trace where they go. You could possibly put some dye in one of the grey water tanks (ie: the kitchen sink drain or bathroom sink or shower drain) to figure out where it comes from. The black tank is your toilet drain. It always has the larger diameter pipe going to the valve (usually 2 or 3 inch diameter) The grey water tubing is usually 1 1/2 inch tubing.

U SHOULD HAVE A BLACK TANK AND A GRAY. TOWARDS THE BACK WERE U HOOK UP THE HOSE FOR DUMPING U WILL SEE TWO LEVERS ONE IS FOR BLACK ONE IS FOR GRAY. AND IF UR TALKING ABOUT FRESH HOLDING TANK LOOK UNDER NEATH AND U SHOULD SEE A PLASTIC NOSEL WITH A TURN VALVE LIKE WHAT BOX WINE HAS TO PURE A GLASS

If it is a big valve it is your sewer tank drain valve. There is two valves. One is the Black water drain and the other is the gray water drain. the black water drain comes from the toilet holding tank, and the gray water valve drains the shower and sink holding tank.

You have the question we are all asking.
rv tank gauges never seem to be very accurate. The best you can do for your black water tank is to try to see where the sensors are mounted and try to spray water on them to loosen any gunk.
I have had 4 motor homes and never had very accurate tank indicators.
Your black tank can be checked by taking a flashlight with the water off and looking down the hole of the toilet. When it gets close to the lowest part,of the drain hole it is time to dump.

On some later model Montana 5th wheels, there are actually three holding tanks which will eventually require draining. (two grey tanks, 1 for the kitchen sink(s) and 1 for the bath tub & bathroom sink, and one black tank, obviously the toilet...) The two most used drain valves (1 black & 1 grey) are together in the docking station area where the city water supply is hooked up, the other (kitchen sink) is at the left rear of the coach behind the farthest rear wheel along the frame. (That is where "Keystone" installed the grey tank for the galley, or kitchen if you prefer.)
If certain it is the black tank which won't dump when the black tank valve is pulled, there is a restriction before the valve (or the valve itself is not opening hence a broken cable to the valve?) One might try "snaking" up into the sewer line, measuring the distance from the outlet to the valve location, assuring it is either open or closed. Once that has been determined, you will at that point have to remove the sewer line at the valve for repair. Be sure to wear latex gloves and eye protection and have a drain pan or bucket handy for spillage which can be collected and dumped down your home toilet or outside sewer clean out if you know its location.

Should be readily visible underneath the unit. Begin your search near the Black Water tank, which is the one most directly connected to the sewage release valve where you hook the dump hose. Usually, Grey Water dumps through the same hose via a second dump valve.