It could be that you have multiple issues. Just recently we had a great series of posts that helped me diagnose my fuel gauge and sender problems.
Gas Gage Issue In my case I had 2 problems. The main issue was that the cork floats were saturated and were sinking to the bottom of the tank. The other point I corrected was to make sure that the moving arm was properly positioned on the variable resistor in the sending unit.
I put in new cork floats and things work pretty good for now. I do see a lot of needle movement both up and down as a I drive. I expect that within a year of so I will be back in there figuring out on to adapt a brass float to the sending unit.
I agree that you are probably going to have to pull the sending unit out of the tank to get things operating correctly. Thank goodness that is easy to do on my coupe.
Like Bloo suggested do the simple tests first before you drop the tank and pull the unit. Ground the terminal where the wire attaches to the sending unit and make sure the gauge reads "E". Then disconnect that wire and make sure it reads "F".
After you remove the unit reconnect the wire to the gauge (you might need a short jumper wire) and also ground the sending unit. Then manually move the float arm through the full range while someone watches the fuel gauge. That will tell you how resistance is changing as the arm moves.
That was the test that helped me figure out that the floats were not "floating". I could get a full and correct range of gauge readings by moving the arm. When I put the sending unit back in the tank it would read the correct level for about a day. Then all it did was read empty. The floats would dry out while I had them out but it only took a day for them to become saturated again.
Sorry for the long post but I know you want to be sure it is the sending unit you go to all the work to pull the tank. And that you can fix it after you get it out.