The thing with thick grease is, it gets pushed out of the way and doesn't return. You could very well have a box full of grease and still have the gear teeth going metal on metal, and it's even worse if the gear box has a line of roller ball that run on the worm shaft.
If you pull the side cover off the gear box and carefully dig out what you can, you can make room for room gear oil. You then can reinstall the cover without changing the adjustments. At most you might have to cut a new gasket. Add some gear oil and work the steering back and forth. Your reddish brown grease is likely contaminated with rust, giving it the reddish brown color. At the lowest point in the gear box is the lower worm shaft bearing, most were barrel shaped and hard to come by, this is where water sets. I had to throw away two worm shafts before I found a good one due to crappy rusty grease.
