"Some in my local club use grease just for this reason..."
If they mean chassis grease, I strongly recommend that you seek future Model A repair advice elsewhere. Anyone who recommends using chassis grease in Model A steering boxes instead of a semi-fluid lubricant or a dedicated steering grease (such as Penrite) cannot be considered a wealth of Model A knowledge or experience. Poor advice is trump in our hobby. Pumping grease into our steering boxes is what pays for parts vendors' kids' college educations by selling new sector gears ($$$$$$$) and worm gears ($$$) to replace badly worn ones caused by lack of lubrication between the teeth. Even if the steering box is crammed chock full of chassis grease, after a few turns, the grease is squeezed out from between the sector gear teeth and the worm gear - and it doesn't magically work its way back in. It cannot coat and remain on surfaces like dedicated steering box lube, which is unbelievably slimy and clingy. How can a near solid material like chassis grease possibly work its way back to lubricate the moving parts, or even the bearings, especially the upper one? Ford put a zerk fitting for a time on top of the steering boxes, but it was never intended for chassis grease. Dealerships during the era had pressurized steering gear lube guns to force the proper lube inside. Because of misuse or misunderstandings in the real world, these zerk fittings were replaced at the factory by plugs so that chassis grease could no longer be pumped inside.
Follow the suggestions posted here so far, but also plan on replacing the sector bushings and then honing them. Otherwise, 600w or equivalent lube will work its way through the too wide clearance and leak out of the side, eventually ending up on your garage floor or driveway. (Yuch!) Lube such as 600w usually leaks out more between a worn sector shaft and its bushings than out of the bottom. Check the inside and outside of the frame where the steering box is mounted. It will most likely be all gooey from leakage past the sector gear shaft and bushings. Only one cure at this juncture: new bushings at the minimum, sized to encase the sector shaft to attain the proper clearance. If the sector gear teeth are gouged, you'll need to replace the sector, too, or you'll never be able to adjust out steering slop. And use a dedicated steering box lube! Penrite is admittedly expensive @ about $19.00 per bottle plus shipping, but it will not leak like 600w and will constantly coat and lube the moving parts. You could probably fill three or four Model A steering boxes with one bottle of Penrite, so the expense could be shared among club members. The right solution is never the cheapest one.
Marshall